Everyone's reaction is different, and they may find different specific details of the situation the most disturbing. My mother and my grandmother have gotten into severe accidents, which required overnight hospitalization. My grandmother was hit about 30 years ago. To this day, my grandmother can't be in a car if anyone but her is driving, although she was the one driving when she had the accident. She's had many panic attacks behind the wheel since then.
I was in a small car accident while I was driving (someone changed lanes into me), and two other small accidents as a passenger. After one of these, I wore a neckbrace for a month. I was also a passenger in an incident where the car spun 720+ degrees and fell off the road between trees, but no one was harmed. Despite the fact that I had worse accidents as a passenger, the accidents for which I was the passenger left me with no anxieties. However the one where I was driving left me uneasy about driving a car for two days, although it was not my fault. I could, but I didn't want to. The day of the accident, I was especially shaky. Fully recovering, to the point that I had no anxiety while driving, took about a month, but obviously something like that varies from person to person over different situations. For me, time was all that was needed to rid me of the anxiety.
External factors and stress, I believe, can influence your reaction. Right before that one woman bashed into me while I was driving, I was driving home from the hospital after being told I'd have a surgical complication (I was having surgery 2 weeks later). It was a stressful time for me, so I reacted more than I usually would have because I was already overwhelmed.
I think certain personalities can lend to anxiety after car crashes too. For instance, I like to feel in control of my life and have already had control taken away from me before, especially through illness. Thus, I probably find it more disconcerting to lack control in such a situation, but this has nothing to do with overreacting in my opinion. It's just how we're wired. In short, don't feel bad about your reaction. It can be a result of many factors, and even if it weren't, it's still nothing to be ashamed of.