Ah I remember when I was 1h cav. It hurt me on a deep, personal level. I did far better when I picked up a lance, despite my 0 wpf in polearms.
Effective heavy cav, the kind that does close infantry support, requires a metal horse of some kind, or a destrier. Regardless, its amazingly expensive if you actually desire to play your role, and its not easy. Do not expect to stand up to people one on one and actually win, for they'll just kill your horse. Or they'll stand infront of it, blocking it, so that you cannot move, despite the 500 pound animal being entirely at your command. Or they'll waggle a polearm infront of its face, bringing your movement to a dead halt.
In other words, have fun being 1h cav.