It might not be immediately obvious, but in the Street Fighter games, you're participating in a tournament. A tournament where you can (more or less) control the outcome.
When your opponent is chosen, press Start on Controller 2 to bring in a New Challenger.
Have Controller 2 pick the character you were fighting against, and pummel them mercilessly. Win the match, and you'll notice that you continue to the next stage.
And since you know you'll be fighting the four Boss Characters, you can also do this to beat them early.
Well, except for M. Bison. You have to beat him the old fashioned way, I'm afraid.
Oh, and this works for all variants of Street Fighter 2.