In our volunteer FD, I've attended more horse accidents than auto accidents. Some of them have had nearly life-ending injuries: flailed chest, book fracture, skull fracture. The rider was a world-class polo rider, and had been riding horses since he was four years old.

In one case, a life-long rancher (that is, a young man raised on a ranch) was standing in the wrong place outside a gate when he opened it, and five horses trampled him, crushing his skull. He died in the ER. 

Those two incidents are the worst cases I've seen. I've seen plenty of other issues: broken collarbones (nb the plural), broken legs, broken ribs. I know one young rider who was in a coma for three months after getting caught in the stirrup and being dragged beneath the spooked horse. Oh, let's see: broken arms, a broken pelvis (but not a book fracture), fingers stumped off when throwing a dally and getting a finger between the rope and the horn... a shattered hand from a horsekick. Give me a few more minutes and I'm sure I'd think of more.

Get your kids a rocket-powered ATV if you care about their safety.