Well, they did overthrow said ruling family remember. And don't forget the Mad Kings which were a direct result of the incest. I think the other powerful families had finally had enough of the mad Targaryens.

Oh, and don't forget the dragons. Intimidating creatures those dragons. Even if every other family outside the Tar's hated them, until the dragons were gone there was not much to be done about it. You'd just end up as a snack for Belarion.

So once the dragons were gone, and mad kings continued being mad....well, death by rebellion. Good ridance Targaryens.