Not really. A horse is typically 'popped' with a whip on the hindquarters, and sometimes the shoulder. These areas are all heavily muscled and not easily wounded by a light pop from a whip (though it is highly possible for welts to be raised). Horses have a higher tolerance to pain; remember horses will kick each other full force and be able to walk it off.
The whip provides more of a sound and a tiny 'sting' from the popper end, nothing more annoying than a fly bite really. Racehorses are trained to respond to the whip from a young age, it's more about the horse understanding that the touch of a whip means to go faster or run straighter, not about inflicting pain on the horse.