The best racing games like Road Rash are Road Redemption, Burnout 3: Takedown and Wreckfest. Below are 9, each with the reason it is here — and, at the end, what filtering to racing costs you.
Filtering to racing means you want the vehicle to be the subject. It is the right filter when what you liked was speed, or handling, or a road worth driving.
What you get: A short session that is complete in itself. A race is over in five minutes and you can always take the corner better.
What made Road Rash worth playing in the first place is covered on the full list. This page is the racing part of that answer, and only the racing part.
Made specifically to be Road Rash again, and it plays like it.
The best combat racer ever made, in cars instead of bikes.
Contact racing where the damage is the point.
Weapons during a race, and a rewind mechanic.
Vehicle combat, if the fighting mattered more than the racing.
Off-road arcade racing with genuine contact.
Bikes, physics and short punishing runs.
Top-down combat racing with the same cheerful nastiness.
For the open-road tone rather than the fighting.
Everything outside the vehicle. Racing games rarely have a story, and the ones that try usually should not have.
The full list is on games like Road Rash — 9 picks, no genre filter.
Road Redemption. Made specifically to be Road Rash again, and it plays like it.
No. Road Rash released on the Sega Genesis, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Simulation models tyre temperature, fuel and weight transfer and punishes you for ignoring them. Arcade gives you grip, boost and a rewind, and is about flow. Buying the wrong one is where most disappointment comes from.
For the simulation picks it changes the game completely. For anything arcade it makes almost no difference.
By hand. Each pick was written for the full Road Rash list first, on what the game actually does, and this page is the part of that list that is racing — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the racing guide for where to start in the genre generally.