It's always been presumed Kirk was on the fast track. After all, in "Court Martial" he's talking to his fellow "classmates" and they're all lieutenants or lieutenant commanders. (I think one is even still an ensign...). Of course, in reality, this is probably just because those were the extra uniforms the costume department had on hand.
It was pretty well implied, but not explicitly stated, that at the time of TOS, Kirk was one of the youngest captains to ever command a starship, at the age of 31.
I have seen the darkness in my soul and shine brighter for it.