I am not sure what a bridge style camera is, this is the first I have heard that term.

I carried a Pentax X90 superzoom type camera on several bike trips in the handlebar bag, no problem. But on one fully supported trip where the tour company provided the bikes with small handlebar bags, the bikes had skinny high pressure tires and when I hit the cobble stones in some of the smaller European villages, I was quite concerned about my camera and vibration. I had a nylon webbing strap, about 3 to 4 feet long and a small drybag about 5 liters. After the first day I started to carry the camera in the drybag and used the nylon strap as a shoulder strap. That way it was not exposed to the vibration.

That said, a handlebar bag usually has enough flexibility in the mounting, that it is pretty effective at reducing shock and vibration from the contents as long as your road surface is reasonably smooth. Thus, I usually just carried the camera in the handlebar bag.

For my last two tours I left my better cameras at home and only carried a Pentax WG-3 camera. It is a point and shoot, waterproof, labeled as shockproof - crush proof - cold proof, and <insert chuckle here> labeled as "adventure proof." I stopped worrying about camera damage from being in a handlebar bag.

The WG-3 is no longer sold under the Pentax name, now sold as the Ricoh WG-4 or WG-5. At only a 4X zoom, not very good for wild life but otherwise serves all my bike touring needs. And I certainly like the waterproof feature.