Notice there didn't seem to be any consensus on a set of rules so I knocked up a quick set for you. Feel free to use, modify or find something you like better.
Each turn a chariot can accelerate or deccelerate one speed point. Then move that many spaces forward. The chariot in front moves first, then the one in second place etc etc.
Drifting - a chariot can drift one space left or right each turn. If they do, make a handling check.
Turns - if a chariot spends at least part of it's move on a curve, make a handling check.
Emergency brakes - If a chariot would run into the back of another chariot they must jam on the brakes. Make a handling check at the current speed and then reduce their speed to the minimum required to prevent a crash.
Side-swipes - if at any point you move alongside another chariot, or if another chariot moves alongside you, you may side-swipe them; both chariots must make a handling check.
Whip - if at any point you move alongside another chariot, or if another chariot moves alongside you, you may whip them; they suffer one point of damage.
Handling checks - to make a handling check roll 1d6. If the result is equal to or more than your speed you have passed with no ill effects. Otherwise suffer one point of damage for each point you failed by. For example, Ben Hur is attempting to take a bend at a speed of 6. He rolls a die and scores a 1. He therefore suffers 5 points of damage!
A chariot has 9 points of endurance. These points are reduced by damage. A chariot with no endurance left must roll on the following table:
1. Spectacular explosion. Remove the chariot. Every chariot within 2 spaces must make a handling check.
2. Crash! Continue moving the chariot next turn (with no side-swipes etc allowed) and then leave the chariot where it is as a wreck acting as an obstacle for the rest of the race.
3. Horse breaks free. Immediately halve your current speed, rounding down. Minus 2 from all future handling check rolls. Move as normal next turn then roll again. If you lose all your horses then chariot stops where it is as an obstacle for the rest of the race.
4. Swerving wildly. The player to your left moves your chariot next turn (No handling checks required). Roll again next turn.
5. Get it together! Reduce speed by one. May not make any side-swipes or other attacks next turn. Roll again next turn.
6. Hang in there! Return to 1 point of endurance.
If a chariot stops dead (and is not a wreck) one point of endurance can be recovered each turn spent stationary.
The game can be made more interesting with a choice of Upgrades, maybe a deck of cunning plan cards or some random events.