I think, from my horse riding history (which I did when I was 7), there is like.. show (?) and then there is jumps racing.
Show is jumping horses over a few obstacles and getting scored for form (like you see in the olympics).
Jumps racing is a race -- going as fast as possible into jumps.
Both are cases of animal exploitation, but jumps racing is far worse than 'show'.
... and then there is the steeple chase/cross counrty (?). It's also very dangerous - going at speed into jumps but landing in rivers and stuff. I think though it is marginally safer than jumps racing.
Correct me if I'm wrong on any of this.

yeah its called show jumping ... Its an olympic event, by its self, or in the 3 day event. 3 day event is show jumping, cross country and dressage (that prancy dancey thing raven mentioned.)
in show jumping they use older horses, (when i say older, i mean like there not 3 like race horses.) and they tend to use breeds that excell at jumping (warmbloods, hanoverians, oldenburgs ect) . Training for these horses can take years.
i wouldnt compare steeple chasing with cross country though. Like there just so different. Once again these horses spend years being train, so when they get to eventing stage their ready. Its much easier for the horse to concentrate on the jump when he's not surrounded by 20 other terrified horses.
What are your thoughts on harness racing?? i think thats way crueller then eventing.