Sounds pretty hazardous (and almost definitely illegal) to me
Here's the first photo of the first suspension bridge I found when I looked on Wikipedia
Can you see all the really narrow guy wires below the bridge? No? I can't either. If you can't see them in a photo, how are you going to see them at 100 knots?
"But Steve, maybe there aren't any!"
Yes, or maybe there are. Are you confident enough to bet your life on it?
I thought you were the one with the plan!
Each pilot in command shall, before beginning a flight, become
familiar with all available information concerning that flight. (FAR 91.103)
Suspension bridges aren't like TFRs, they don't just appear in between the time you took off and the time you get to them. Why didn't you know there was a bridge there before you took off? Why didn't you have a plan for what to do when you got there already?
Boats are people too
You need to stay 500 feet from any structure. Bridges are structures, so you need to be 500 feet away from them. You also need to be 500 feet away from vessels and people, so if there's anyone on the river, your options just went from "bad" to "none".
But what about my emeeeergency?
All the rules go out the window if you're experiencing an emergency...as long as you're acting in the interests of the safety of the flight. I spent some time on a foggy morning talking with an FAA inspector about this very topic (flying under bridges in emergencies) a couple years ago, and his words were to the effect of:
"It would have to be a very unique emergency for flying under a
bridge to be the safest course of action."