Other mistake: When Dr. Strange rushes a patient to brain surgery they show him and Dr. Christine Palmer operating without surgical masks. When they cut away and come back they're wearing masks. Also, Dr. Palmer is an ER doctor, and her assisting him would never happen in real life.
Other mistake: After the fight where his sling ring is stolen by Kaecilius, he has a sling ring in his possession and uses it to jump to Hong Kong.
Chosen answer: The airbag did deploy. However, the deployment occurred when the car first crashed and we see the accident from the outside, so we can't see it. Once we get a shot of the interior, you can see the airbag is sticking out from the steering wheel, but (unlike what some movies would show you) airbags deflate very rapidly after a crash, so you may have missed it.
But if the airbags did deploy, then why did he have such major injuries? Wouldn't the airbags have made what happened to his hands less severe?
I don't think so. I watched it again recently and I think I saw the car's speedometer crush his hands while they were still on the steering wheel, while the airbag deploys from the centre of the steering wheel, nowhere near his hands.
Often in real life, when air bags deploy from the steering wheel, they force the hands off the steering wheel resulting in the hands hitting the windshield. Damage to hands, wrists, and lower arms are common.