I don't know about the National Hunt - does that involve jumping fences, plus whatever else?
Yes, there are two branches, hurdles and steeplechases. In a hurdles race, the horses jump over obstacles called hurdles, in a steeplechase they jump plain fences, water jumps or open ditches. One of our biggest National Hunt races is the Grand National. There was a horse called Red Rum who achieved rock star status in the UK after winning it 3 times - 1973, 1974, 1977. He came second in the intervening years! The Grand National is a notoriously difficult race that has been described as "the ultimate test of a horse’s courage". He was also renowned for his jumping ability, having not fallen in 100 races. The 1973 race in which Red Rum secured his comeback victory from 30 lengths behind is often considered one of the greatest Grand Nationals in history.
Another much loved horse was Desert Orchid, who was much loved by supporters for his front-running attacking style, iron will and extreme versatility.