No, there is Arabian, Appaloosa, Thoroughbred, Quarter Horse, American Paint Horses, and mutt races. I know this because I am a horse racing freak. I keep a log on all TB race horses that have a chance in the big races (Kentucky Derby, Melbourne Cup, Belmont Stakes, etc...)
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Yes the Thoroughbred is a purebred. The Thoroughbred has been a purebred ever since it's studbook closed to outside blood.
The quarter horse is the fastest horse over the quarter of a mile, but the thoroughbred is faster over larger distances.