As a matter of fact, this is an interesting question as the car had several effects on horse populations here in the US:
1) Horses became bred for their other abilities. Horses were and still are readily used on farms and rural properties, though not as frequently as before. There are still areas more reachable on foot or horseback than by car or flight.
2) The car grew to popularity across the country in larger cities first, thus giving the "horse and buggy" or carriage form of transportation an opportunity to die down in popularity and the breeding of draft horses to die out a little at a time.
3) Horses were also set free by owners who could no longer afford to feed or keep the horses given their lack of value. This happened across the country and resulted in many stray horse colonies that have cropped up from the remote Cumberland Island to the Cimarron areas of the Western United States. Populations have died out due to overcrowding and natural selection (many of these horses were not bred to survive alone and were intended to be domesticated), but they still exist.
4) Horses became known for other novelty culture activities which rose to popularity around the early 1900's, namely racing, showing, and in hunting parties for sport.
Over the decades it took for this transformation to happen, there was still some overburden of horses, but the transition was long and often cruel to the animals. In the end, it appeared smooth but the effects on the economy were devastating for many who had staked their claim in the draft horse business.
I hope this helps!