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.
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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.
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And since you know you'll be fighting the four Boss Characters, you can also do this to beat them early.
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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.