Well actually their were races where mixed breeds of quarter horse and thoroughbred raced, but those horses were pushed so hard that their hearts actually exploded and now its illegal so I guess you are right, but Thoroughbreds are by far the most common racing horses.
Actually if we are talking top speed the quarter horse is faster, as the fastest quarter horse was clocked at 55 mph, while the fastest thoroughbred clocked was 44 mph. Quarter horses are the fastest horse in a quarter mile "hence" the name quarter horse. Thoroughbreds are more used for racing as they can remain at a top speed for a longer distance.
But Arabians, or Mustangs (which descend from Arabian stock) are the fastest over many miles.