The Mozart movie convinced the casual viewer that everyone knew his songs by playing Nachtmusik. The only Mozart song everyone knows. Not even Turkish march is ubiquitous, Nachmusik is the only one.
There is a website with a Mozart-or-Salieri game where you have to listen to other Mozart clips, and Salieri clips, and try to guess which one of them composed it. It's damn near impossible, it all sounds the same, and 99% of it is boring light melodical fluff. Strings, flutes, stuff people would listen to while doing homework. It was garbage, nobody likes it anymore.
Beethoven had 4 hits: 9th, Moonlight, Elise, and the intro to the 3rd. That's all. Moonlight was the equivalent of Eddie Van Halen Eruption, a skill piece with no musical value at all. Elise is for kids to play at recitals. 9th is good.
Bach is famous for the D Minor fugue, which noone can prove he actually wrote, and is unlike anything he is known to have written. Noone would even remember that if it weren't for Dracula movies.