TL;DR: Discriminatory policies against Jews came long before the death camps, and the US was well aware of these policies by the mid-30's. The US government became aware of internment of Jews soon after that policy was enacted as well. The existence of death camps came to the attention of the Allied governments by 1942 at the latest, but wasn't fully believed at first. By the time of the Normandy landings, though, many average citizens of the US - especially American Jews - were well aware of the rumors about such facilities.
Before the holocaust:
The US was aware of the Nazis' oppression and brutality towards the Jews soon after Hitler took power.
In 1933, new German laws forced Jews out of their civil service jobs, university and law court positions, and other areas of public life. In April 1933, laws proclaimed at Nuremberg made Jews second-class citizens. These Nuremberg Laws defined Jews, not by their religion or by how they wanted to identify themselves, but by the religious affiliation of their grandparents. Between 1937 and 1939, new anti-Jewish regulations segregated Jews further and made daily life very difficult for them. Jews could not attend public schools; go to theaters, cinema, or vacation resorts; or reside or even walk in certain sections of German cities.
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The US, and the other future Allied powers, did little to oppose the early signs of anti-Jewish policies in Nazi Germany.
In 1933 the Nazi Party under the leadership of Adolf Hitler legally gained power in
Germany, and it did not take long for the persecution of Jews to begin. On April 7, 1933,
the Reichstag, under Hitler’s influence, adopted the Restoration of the Professional Civil
Service Law that dismissed non-Aryans from governmental positions.
This restoration
of this act was more significant than it appeared because the Civil Service decree
included bankers, lawyers, railroad and hospital workers, medicine, law, schools,
universities, and the arts. The Nazi intent of these laws was to eliminate Jewish influence
in Germany.
Not long after the Nazis implemented the Civil Service laws in Germany the first
significant protest against Jewish mistreatment came from the American Jewish Committee. This was a committee developed to protect the rights of Jews, and they
asked the American government to properly investigate what was happening in
Germany.
The American Jewish Congress began protests and had a strong membership
and following that would help inform the rest of the country of the worsening conditions
in Germany. For the United States to protest another country’s treatment of its citizens
during the era of segregationist Jim Crow laws was somewhat hypocritical. The United
States had no basis to criticize another country for racial injustice when it practiced
similar discrimination of certain races of people. Nonetheless protests began and
Christian groups, such as the American Christians, made public protests against Nazi
anti-Semitic acts with the support of influential public figures such as Alfred E. Smith,
Newton D. Baker, and John W. Davis.
Slowly information about Nazi anti-Semitic acts
began to filter into the United States. The State Department after hearing of these
allegations asked the American Embassy in Berlin to investigate these accusations. The
embassy liaison reported that there was good indication that the Nazis were hiding
something.
When the Germans were questioned on this issue of Jewish mistreatment,
the Nazis claimed this anti-Semitic rhetoric and attacks were isolated incidents and they
were in process of being stopped. This was very early in Hitler’s regime and the true
magnitude of his hatred of Jews was not yet identified. Ranking United States
government individuals probably knew that Hitler inherently disliked Jews because of his
biography Mein Kamp, but there was no precedent of Jewish genocide to use as a guide
to future events.
Throughout 1933 reports of Jewish mistreatment at the hands of the Nazis kept
surfacing. H.R. Knickerbocker, who was the New York Evening Post correspondent in
Berlin, reported in April 1933 that an undetermined number of Jews had been killed, or
fled, or been deprived of their livelihood in the Reich.
In this Newsweek article Mr.
Knickerbocker made two interesting observations. The first was that he did not
understand why the Nazis had such hatred for the Jews, and secondly he concluded that
the Germans were envious of Jewish accomplishments in Germany. These two conclusions would prove to be very astute in the years that followed, yet few people in
the United States seemed to understand what was happening to the Jews in Germany.
From the everyday citizen to the higher powers in the government this hatred was not
fully understood until it was too late. The Nazis under Hitler’s leadership ignited the
hidden anti-Jewish attitudes of gentiles in Germany. Hitler was a master orator and he
convinced the German citizens that the Jews were the reason that Germany lost World
War I and fell into economic despair in the 1920s.
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The easiest way for other countries to help the Jews of Germany was to increase immigration quotas, but this was not politically feasible for many years, because the world - including the US - remained mired in the Great Depression, and public sentiment was focused on finding jobs for American citizens at the expense of all other concerns. Furthermore, Anti-Semitism was rampant across Europe and the US at the time, so sympathy for the Jews of Germany was fairly hard to find.
Many German and Austrian Jews tried to go to the United States but could not obtain the visas needed to enter. Even though news of the violent pogroms of November 1938 was widely reported, Americans remained reluctant to welcome Jewish refugees. In the midst of the Great Depression, many Americans believed that refugees would compete with them for jobs and overburden social programs set up to assist the needy.
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And:
Like most other countries, the United States did not welcome Jewish refugees from Europe. In 1939, 83% of Americans were opposed to the admission of refugees.3 In the midst of the Great Depression, many feared the burden that immigrants could place on the nation’s economy; refugees, who in most cases were prevented from bringing any money or assets with them, were an even greater cause for concern. Indeed, as early as 1930, President Herbert Hoover reinterpreted immigration legislation barring those “likely to become a public charge” to include even those immigrants who were capable of working, reasoning that high unemployment would make it impossible for immigrants to find jobs.
While economic concerns certainly played a role in Americans’ attitudes toward immigration, so too did feelings of fear, mistrust, and even hatred of those who were different. Immigration policies were shaped by fears of communist infiltrators and Nazi spies. Antisemitism also played an important role in public opinion.
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For these reasons, many historians rightly assert that other European nations, and the US, share some of the blame for the holocaust.
During the holocaust:
By 1940, reports on Auschwitz in particular were arriving in London, via the Polish resistance and the Polish Government-in-Exile.
In 1942, the Polish Government-in-Exile submitted a paper describing the mass extermination of Jews in occupied Poland. It was not the first such paper they had produced, but it was the first to be made public.
Note to the Governments of the United Nations - December 10th, 1942
Ponizszy dokument informuje wszystkie Narody Zjednoczone o exterminacji Zydow polskich i europejskich przez Hitlerowcow na terenach okupowanych.
INTRODUCTORY NOTE
The purpose of this publication is to make public the contents of the Note of December 10th, 1942, addressed by the Polish Government to the Governments of the United Nations concerning the mass extermination of Jews in the Polish territories occupied by Germany, and also other documents treating on the same subject.
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A week later, the New York Times covered the paper:
ALLIES CONDEMN NAZI WAR ON JEWS; United Nations Issue Joint Declaration of Protest on 'Cold-Blooded Extermination' 11 ALLIES CONDEMN NAZI WAR ON JEWS
December 18, 1942
WASHINGTON, Dec. 17 -- A joint declaration by members of the United Nations was issued today condemning Germany's "bestial policy of cold-blooded extermination" of Jews and declaring that "such events can only strengthen the resolve of all freedom-loving peoples to overthrow the barbarous Hitlerite tyranny."
- NY Times
Here is the full text of the Joint Declaration by the UN:
"The attention of the Belgian, Czechoslovak, Greek, Jugoslav, Luxembourg, Netherlands, Norwegian, Polish, Soviet, United Kingdom and United States Governments and also of the French National Committee has been drawn to numerous reports from Europe that the German authorities, not content with denying to persons of Jewish race in all the territories over which their barbarous rule has been extended, the most elementary human rights, are now carrying into effect Hitler's oft-repeated intention to exterminate the Jewish people in Europe.
From all the occupied countries Jews are being transported in conditions of appalling horror and brutality to Eastern Europe. In Poland, which has been made the principal Nazi slaughterhouse, the ghettos established by the German invader are being systematically emptied of all Jews except a few highly skilled workers required for war industries. None of those taken away are ever heard of again. The able-bodied are slowly worked to death in labor camps. The infirm are left to die of exposure and starvation or are deliberately massacred in mass executions. The number of victims of these bloody cruelties is reckoned in many hundreds of thousands of entirely innocent men, women and children.
The above-mentioned governments and the French National Committee condemn in the strongest possible terms this bestial policy of cold-blooded extermination. They declare that such events can only strengthen the resolve of all freedom-loving peoples to overthrow the barbarous Hitlerite tyranny. They reaffirm their solemn resolution to insure that those responsible for these crimes shall not escape retribution, and to press on with the necessary practical measures to this end."
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In May of 1943, an Allied conference in Bermuda decided not to increase immigration quotas in the US and UK for European Jews fleeing Nazi occupation, effectively condemning many people to death in the camps. The next week, the American Zionist Committee for a Jewish Army published a piece in the New York Times, denouncing the decision:

By this point, the Allies' initial reaction - dismissing the reports of mass extermination as Polish propaganda - was untenable. For the past year, the Allied intelligence services had been intercepting and decoding German police communiques detailing intake and death records from many concentration camps. These intercepted messages are known as the Bletchley Park Concentration Camp Decodes.
The cat being out of the bag, the mainstream press began reporting on Nazi policies of mass extermination more frequently after 1943.
Conclusion:
It would have been possible for an average citizen/soldier from the US to remain mostly unaware of the holocaust, but almost everyone would have been aware of Nazi persecution of Jews by 1944, and many would have heard at least some rumors or reports of mass extermination. American Jews would have been much more acutely aware of these stories, and would be likely to treat the Nazis as their enemies on a deeply personal level - much more so than gentile Americans, who were more opposed to Japan than Germany, did.