1889: I think this must be the Fulham Railway Bridge, in which case the date must be at least 1889 (and not about 1878 as is said) -

Fulham Railway Bridge, 1889?
Fulham Railway Bridge, 1889?

1880: A railway line was built by the London & South Western Railway as far as the north end of the present bridge.
1889: Fulham Railway Bridge was built. 750 feet long. Five turquoise wrought-iron lattice girders supported by pairs of cast-iron cylinders provide two railway tracks for the District Line to Wimbledon. Engineer: William Jacomb. Contractor: Head Wrightson

1897: Putney Railway Bridge, James Dredge -

'Putney' Railway Bridge, James Dredge, 1897
'Putney' Railway Bridge, James Dredge, 1897
© Oxfordshire County Council Photographic Archive; D230197a

It currently carries the District Underground line. It is now chiefly used by commuters and when it was built the area around Fulham was still mostly countryside and was popular with day trippers from other parts of London who enjoyed the fresh air and the many boat trips available.

1997:  Strengthening -

Fulham Railway Bridge 1997
Fulham Railway Bridge 1997

Fulham Bridge © 2000 Doug Myers
Fulham Bridge © 2000 Doug Myers

Fulham Bridge PLA
Fulham Bridge going upstream