This was discussed some years ago in the dev forums. I can't remember all the details but - as far as lore is concerned - here's my take:
1. The official maps show a road but no bridge
2. The Atlas (not official) does show a bridge, though where that information comes from, I don't know.
3. Tarannon Falastur built arched columns out into the Anduin from Pelargir during the Third Age (on which he built a house) - could this have become confused with a bridge?
4. Ships travelled up the Anduin to Harlond, and so there could not have been any bridges over the Anduin before that point.
5. Is it not likely the road on the east bank terminated in some sort of port to allow armies to cross over to Pelargir by ship? i.e. the road was built for convenience to access the port and not to access a bridge?
Just some thoughts. If anyone knows better, let us know.