As has been said above, the Severn Bridge was primarily built to take coal traffic to Sharpness from the Forest of Dean. That traffic never materialised in the anticipated volume.
It was begun after the tunnel (in 1875) but finished well before (in 1879).
The bridge was never intended to be a major through route – indeed, it was not really capable of being one, even leaving aside the fact it was single track and the weight restrictions. Traffic from the north for south Wales would naturally go via Gloucester. Traffic from the east for south Wales also had to go via Gloucester because there is no triangular junction at Stonehouse (although I suppose a reversal at Standish Jct might have been possible, if an operational nuisance). Traffic from Bristol for south Wales might in theory have gone via the bridge, but that would have required a reversal at Berkeley Road.
Of course once the tunnel opened, such coal traffic as there was to Sharpness could have gone that way, but until the Badminton Line opened at the start of the twentieth century, that would have necessitated a reversal in Bristol as well as a reversal at Berkeley Road. Even once the Badminton Line had opened, it would have been a rather circuitous route and still required the Berkeley Road reversal.