Here's how I interpret the Mario back story, even although it has NOTHING to do this topic.
Yoshi's adventures were on Yoshi's Island. As we know from this, Bowser's first arch-nemesis or whatever was Yoshi, not Mario (according to ''chronologic'' order). We also know that the map the stork was presumably using to find Mario's family was a map that Yoshi couldn't figure out.
This either means Yoshi can't read maps, or it isn't a map of Yoshi's Island. Doesn't quite explain why the level select screen is a map, though. If it was a map of Yoshi's Island, then the mushroom house might be somewhere on the island (unlikely).
The mushroom house, since we can't identify where it is, is not in the Mushroom Kingdom or Brooklyn. Chances are it's in the Mushroom Kingdom since it's a mushroom house, so this doesn't explain how they got to Brooklyn.
Since Nintendo obviously didn't put together some loose ends when making games (how were they to try to fit SMB with Yoshi's Island a few years later?), they have created a paradox like with baby mario and Mario existing at the same time. This is why no sense can be made of it.
So anyway, they somehow came to Brooklyn. Let's just assume the mushroom house is in Brooklyn (hey, they're Marios, anything can happen). From the cartoons, and the SMB manual, they just so happened to stumble into the Mushroom Kingdom. That pretty much sums up everything, except for how Yoshi and Mario reunite a few years later with Yoshi not remembering a thing about them. Don't give me any of this "There's two Yoshies" stuff, because Nintendo leads us to believe this other story they had.
So in the end, Nintendo created a paradox, so we'll never come to a conclusion. And they just make games, not stories. The whole turtle (yes, Bowser is a turtle, because Koopa Troopas were classified in SMB as the Turtle Tribe, and Bowser is King of the Koopas). capturing princess deal is too overused of a story. And finally, Yoshi is a dinosaur, again according to Nintendo.
That should settle everything. We usually think that what Nintendo says is always correct. So we can't fill in any of these story gaps without completely ignoring Nintendo fact. The cartoons did not fit in with the rest of the game (since when does Madzilla pop up in SMB3?), the prequels and such have loose ends, and that's it.
Now I'd like to hear someone that disagrees with all this, which is probably most of the people that reply to this post.
To be a Koopa, you must become a Koopa.