Brightside Bridge was on a list of bridges in the West Riding compiled for the Quarter Sessions in 1730. The bridge and approaches were "adopted" by the Corporation in 1857.
The following may help:
A Rough Draught of that part of [Car]Brook [Carbrook] Estate contiguous to the Wheels and Tilting Mill Date: 1741
Surveyor: William Fairbank I.
Fields, etc. between the Don and the Carr Brook, with perspective sketches of Brightside Bridge (three arched), a farmhouse and the wheels and tilt; an historical account of the development of the wheels is given, with an explanation of the causes of flooding; flooded areas marked in with acreages. (Carbrook Street) Brightside Forge and Nether Forge, including part of the Carbrook Estate. Original at Sheffield City Archives: ACM/MAPS/SheD/786S