Kaecilius: You'll die defending this world, Mister...
Dr. Stephen Strange: Doctor.
Kaecilius: Mister Doctor?
Dr. Stephen Strange: It's Strange.
Kaecilius: Maybe. Who am I to judge?
Dr. Stephen Strange: This doesn't make any sense.
The Ancient One: Not everything does. Not everything has to.
The Ancient One: We never lose our demons, we only learn to live above them.
The Ancient One: You think you know how the world works? You think that this material universe is all there is? What is real? What mysteries lie beyond the reach of your senses? At the root of existence, mind and matter meet. Thoughts form reality. This universe is only one of an infinite number. Worlds without end; some benevolent and life-giving, others filled with malice and hunger. Dark places where powers older than time lie, ravenous and waiting. Who are you in this vast multiverse, Mr. Strange?
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.