It seems almost impossible to judge thousands of people per city in such a short time. Do the judges judge everyone or just select people from a crowd?
Contestants go through at least three sets of cuts. The first is a brief audition with a few other contestants in front of selectors which may include one of the show's producers. Although auditions can exceed 10,000 in each city, only a few hundred of these make it past the preliminary round of auditions. Successful auditioners then sing in front of producers where more may be cut. Only then can they proceed to audition in front of the judges, which is the only audition stage shown on television. Those selected by the judges are sent to Hollywood. Between 10–60 people in each city may make it to Hollywood.
So, what we see on TV with the celebrity judges is the third step of initial auditions.