As a general rule, you should be able to build tram tracks on bridges. I won’t suggest you to destroy and rebuild the bridge, because if the game doesn’t let you upgrade now, it might not let you rebuild the bridge in the same spot. Try another route or use busses, it’s safer.
Bridges are stupid like that. Sometimes they won’t let you upgrade them (the game treats tram tracks as road upgrades), I still haven’t found out why. Possibly it’s something related to buildings spawning near their ramps and altering the terrain a bit.