From Roader's Digest: The SABRE Wiki
Like Bath Bridge to the east, Bedminster Bridge is in fact two bridges crossing the River Avon, to form a roundabout. The A370 is heading east west, across the North-south A38, but as the A370 splits to run along the north and south banks of the Avon to Bath Bridge, both routes cross the River!
The Western bridge is the older of the two, and looks Victorian in style. It carries 3 lanes of citybound traffic, and also has a partial cycle lane between the left-turn and straight-ahead lanes to assist cyclists. The eastern, southbound bridge is just 3 lanes wide.
The 1963 AA Book of the Road doesn't show the new bridge here, but does show the new bridge at the Bath Bridge junction upstream. It seems likely that the junctions were upgraded at a about the same time, suggesting that the newer bridge here dates from c1964/5.