Just wanted to add to what everyone else has been saying: I rode at a small barn where all of the lesson horses were perfectly trained, and I only once went over the head of one of the ponies, and it was my own fault. I recently moved to a different barn, and many of the lesson horses aren't for beginners. Before I had moved, I was dead set on buying an untrained horse, and teaching him how to jump. I rode a 17.3hh warmblood gelding named morrie, and I experienced first-hand what inexperienced trainers can do to a horse. He had been imported from Europe by a rich family when he was four or five, but none of them were experienced enough to ride him. Now he has some pretty bad habits because they weren't able to correct them.
I've been jumping for six years, and riding dressage for seven. I'm still not ready to train a horse, and I don't think I will be anytime soon. Sure, i've ridden the four-year-old pony, the difficult horses with various issues, and horses like Thumper who just take off after every jump and hollow out, but there are still so many problems that I HAVEN'T experienced first-hand, and I know that if I ever ran into one of those problems, I would have no clue whatsoever how to fix it.
My advice? Buy a horse who already knows how to jump. If you can't afford a trained horse, lease. But if you're dead-set on buying a horse and training them to jump, please please please
get a trainer to help you along the way.