Believe it or not, I actually kind of suck at the Adventure Island games. It's not really my fault, though. Your guy bites it after being hit once by anything, and enemies are strategically placed right in the best place to jump.
It's devious.
So if you get stuck and throw your controller across the room in frustration instead of making progress, then I understand and have some news you might be interested in.
If you go to the title screen, and let the demo start
Then immediately cancel out of it and go back to the title screen. Then press Right on the Control Pad + L + X + Start. Done right you'll be whisked away to the Stage Selection Screen!
Which is great for previewing those later levels before you retire the game completely.