"It now remains for me to give a brief account of the games which the Romans performed after the procession. The first was a race of four-horse chariots, two-horse chariots, and of unyoked horses....In the chariot races two very ancient customs continue to be observed by the Romans....The first relates to the chariot drawn by three horses....For running beside two horses yoked together in the same manner as in the case of a two-horse chariot was a third horse attached by a trace....The other custom is the race run by those who have ridden in the chariots....For after the chariot races are ended, those who have ridden with the charioteers...leap down from their chariots and run a race with one another the length of the stadium. And after the chariot races were over, those who contended in their own persons entered the lists, that is, runners, boxers, and wrestlers....And in the intervals between the contests they observed a custom that was typically Greek and the most commendable of all customs, that of rewarding crowns and proclaiming the honours with which they rewarded their benefactors."
Dionysius of Halicarnassus, Roman Antiquities (VII.63)
There also was the Diversium, in which the winner changed places with his defeated rival and raced the opposing team of horses as a demonstration of his skill. Desultores were acrobatic riders who led a second horse and leaped from one to the other. Curiously, there seem to have been no races in which a jockey actually rode the horse.
"Who had the skill to fashion so many figures out of one block of marble? The chariot melts into the charioteer; the horses with one common accord obey the same reins. These are distinguishable by their various forms but made from one and the same material without distinction. The driver is of one piece with the car: to this are attached the steeds, each joined to, and proceeding out of, another. How admirable the artist's skill! A single block combines with itself all these bodies: one mass of marble by submitting to the chisel has grown into all these various shapes."
Claudian, Carmina Minora VII (Statue of a Chariot)
This magnificent sculpture is in the Sala Della Biga (Vatican Museums), which unfortunately has been closed due to damage caused by visitors. The picture was taken by a sympathetic guard as a much appreciated favor. Unlike the heavy and ornate chariot paraded in the triumph and ceremonial procession, the racing chariot was stable and light, with small wheels and a wooden frame, and would have been less of a burden for the horse to pull than even a rider on its back.
Reference: Claudian: Shorter Poems (1922) translated by Maurice Platnauer (Loeb Classical Library).