This modern bridge has replaced the old railway bridge for the Melton Constable to Norwich City line. This used to be an A-frame bridge before it was removed (all the others were not); the concrete base has been reused. During WW2 a coal train went through the buffers here and into the river. Known as the Wensum dipper.
I think the path you are talking about is Marriott's Way - running from Norwich to Reepham on the course of an old railway. It's part of National Cycle Route 1 (Harwich to Hull).