The effect (or at least a large part of it) you're probably seeing is called rolling shutter.

As for how to fix it, you're probably right about your mount amplifying the vibration. Something that connects the camera more directly to the bars or the frame would likely work better. Getting it off your bars and onto the frame will help considerably, but of course most bikes don't have any good places to put it and have it facing forwards.

Putting it on your helmet would remove the problem as well (as your body would damp out the vibration) but then replace it with your head turning all over the place -- which might actually be a good thing.

Don't worry too much about a crash. If you crash and your helmet hits the ground hard, the fact that your camera was probably destroyed will be the least of your worries. And it probably won't affect the impact much, as it's likely to be broken off before it affects the impact much.

If you're aiming something backwards, I've found that mounting a camera to your seat post tends to make for a pretty good, solid connection that vibrates as little as possible. But some vibration is unavoidable (unless the camera is on you, of course.)