Bonnie and Clyde's fame was actually very short-lived and soon turned into infamy, but during a time of economic sorrow, the public's fascination with this duo, stimulated by the publication of photos of their extravagant lives and the unfolding of their provocative story, gained them considerable notoriety.
In 1933, after a police raid forced them to flee their hideout in Joplin, the police found, among other things,
a handwritten poem by Bonnie, and a camera with several rolls of undeveloped film. Police developed the film at The Joplin Globe and found many photos of [Clyde] Barrow, [Bonnie] Parker, and Jones posing and pointing weapons at one another. The Globe sent the poem and the photos over the newswire, including a photo of Parker clenching a cigar in her teeth and a pistol in her hand. The Barrow Gang subsequently became front-page news throughout America.
The photo of Parker posing with a cigar and a gun became popular. Jeff Guinn, in his book, Go Down Together: The True, Untold Story of Bonnie and Clyde, noted:
[..] the Joplin photos introduced new criminal superstars with the most titillating trademark of all—illicit sex. Clyde Barrow and Bonnie Parker were wild and young, and undoubtedly slept together.
In this Washington Post article, Guinn expands on this:
The photos were given to a local newspaper and used for a wanted poster. Once they hit the wire services — the pre-Internet in many ways — newspapers across the country printed them, as did true-crime magazines.
“They wanted to cram as many Barrow Gang stories into their publications as possible before the cops inevitably caught the kids and ruined a good story line,” Guinn wrote.
This was because
Bonnie and Clyde’s exploits took place during the Great Depression, when people were beleaguered. Newspapers, themselves trying to survive, figured out that readers were tired of stories about the miserable economy.2
The fact that Bonnie and Clyde robbed banks during the Depression's bank panic garnered them some popularity:
Several criminals operating during the Great Depression, including Bonnie and Clyde, became famous as “Robin Hood” figures who struck back against the banks that many considered to be oppressive.3
But the "ten or so"1 banks they robbed were often small rural banks, and their gang actually preferred to rob funeral homes, shops, and gas stations along backwater streets, thus targeting small local businesses.
Eventually, the cold-bloodedness of their murders opened the public's eyes to the reality of their crimes, and led to their ends.
[..] the massive negative publicity increased the public clamor for the extermination of the Barrow Gang.1
In the film, in Coffeyville, Kansas, the car and its inhabitant, having parked behind a drug store, are being recognized by a young woman, who immediately fetches her friends. Soon the place is crowded with young people, a "goddamn fan club", who apparently want their picture taken with the famous couple. I believe this is the scene you refer to in your question.
While Bonnie and Clyde were, like any fad, popular for a while, there is no record of this or similar events happening, and the whole situation seems highly unlikely to me. The Highwaymen makes use of artistic freedom to dramatize the story, and Hamer and Gault never actually ran into Bonnie and Clyde until the very end.
On the 23rd of May, 1934, right after Parker and Barrow had been killed by the six lawmen that had fired around 130 shots (167 according to other sources) at the Ford V8 they were driving, the car wreckage at the side of a country road gathered a crowd:
Nearly everyone had begun collecting souvenirs such as shell casings, slivers of glass from the shattered car windows, and bloody pieces of clothing from the garments of Bonnie and Clyde. One eager man had opened his pocket knife, and was reaching into the car to cut off Clyde's left ear.1
The depiction towards the end of the film of the crowd surrounding the vehicle as it's being towed into the town of Arcadia, and picking at it like vultures would a carcass, is very credible.
But their demise kept attracting attention:
More than 20,000 attended Parker's funeral, and her family had difficulty reaching her gravesite.1
As for their 'Death Car':
Like the legend of Bonnie and Clyde, the car has survived as a grotesque symbol of fame.2
1: Bonnie and Clyde; Wikipedia
2: Bonnie and Clyde were Depression-era Kardashians: A source of public fascination; Washington Post
3: Bonnie and Clyde; Brittanica