When riding in London a few years ago I came off my bike when a car driver - who hadn't seen me - turned across the road in front of me. I slammed on my brakes, and went over the handlebars only a few inches before hitting the car. I landed on the road with my hip and my head whiplashed over into the road. If I hadn't been wearing a helmet I would undoubtedly have fractured my skull, and possible worse. Since then I'…
What kind of brakes do you have? I've got the feeling that aggressive brakes on your front wheel make it more likely for you to go over the handlebars. I've also been cut off once by a car who didn't see me while I was going very fast, but while I braked, I also made a flat turn into the street the car was turning into, and that saved my from a collision.
The rear brake is much more likely to cause a fall. When you apply the brake the bike tips forward and the rear wheel rises up. It easily looses grip and can lock up. That causes a skid and next thing you are on the ground. Or just smashed into whatever you are trying to avoid.
The front brake does not do this because the tipping forward pushes the front wheel into the ground. You get more grip and more braking potential. With a little practice you can stop the bike quickly and easily. You will never go back. But you have to grip the handlebars and tense your arms a bit to deal with the force.