Crossing the bridge in a car is only possible with cheats - the Banshee just flew into the water.
It is possible to get to Staunton before the bridge is fixed without cheats though, but only on foot.
Unless you have completed the ambulance mission it will take a while to get to the end of the bridge though.
Once on Staunton, you can get to Shoreside through the tunnels, as it is open. There's also no traffic. The subway is open as well, as an alternative.
Watch out if you get wasted or busted - you get returned to Portland until legal access is obtained