Like Bath Bridge to the east, Bedminster Bridge is in fact a pair of bridges crossing the River Avon, each part of a roundabout. The A370 is an east-west route, the A38 runs north-south, but with the A370 splitting to run along the north and south banks of the Avon as far as Bath Bridge, both routes cross the river.
The western bridge is the older of the two, and looks Victorian in style. It carries three 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 three 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 roughly the same time, suggesting that the newer bridge here dates from c. 1964-5.