The game suggests repairing your car to avoid performance loss after the race but it's not clear whether it affects it right away.
At least with the current update, the performance does suffer in the race when you damage your car. It seems disconnected from the persistent damage between races.
When damaging your car in a race, acceleration, top speed and grip may suffer. I haven't noticed bad breaking performance yet, but it seems odd that it should be an exception to that rule.
It seems to happen it certain steps (or at least is noticeable in steps). Listen for that hissing sound immediately after a collision (maybe indicating a ruptured hose/pipe?), which seems to indicate that it was hard enough to damage something.
As far as I have experienced, driving off-road does only influence the persistent damage, not the immediate one. The persistent damage is far more influenced by the driven distance and off-road time, while immediate damage results mostly from collisions.
The game's online help says:
I think the wording is quite unambiguous - damage affects your car's performance immediately.
Edit: However, based on my own experience I'm pretty sure that's not entirely true. Yesterday I managed to damage my McLaren MP4-12C badly enough to take the grip rating down to 0.68g and still noticed no noticeable change in the car's handling during the race.