Hard to be definitive - about 10 times at least.
1) there was probably a temporary military bridge put up by the Romans which was destroyed c AD55 to be
2) replaced with permanent timber bridge, destroyed by Bodecia AD 60
3) rebuilt, lasts until Romans abandon London when it eventually disappears during Saxon era
4) rebuilt by Alfred the Great in 9th century destroyed by c1014 probably by Olaf
5) rebuilt by William I after 1066 destroyed in a storm in 1091
6) rebuilt by William II soon after and destroyed by fire 1136
7) rebuilt in timber for the last time and then destroyed by Henry II 1176 for new stone structure
8) rebuilt stone bridge complete c 1209
9) bridge lasts 600 years with a few partial collapses; houses eventually cleared, bridge destroyed after 1830
10) new stone bridge by John Rennie built alongside medieval bridge c 1825, originally designed for horse-drawn traffic, survived the Blitz, destroyed in 1960s as it was subsiding significantly, most of the stone facing was sold off to be re-erected on a new structure at Lake Havasu City as an exhibit - the new alignment is upstream of the medieval bridge, the other side of Adelaide House.
Lord Holford designed present bridge which was opened in 1973