The section of the only crossroads from occupied Crimea collapsed after an explosion.
Widely shared images show a train on the Kerch railway bridge on fire, with the road bridge next to it appearing to be constantly damaged.
Russian state media said a fuel tank was on fire without specifying a cause.
The bridge, built on the orders of Vladimir Putin, is an important link for the transfer of military equipment into Ukraine. Moscow had illegally annexed the region four years earlier.