Around 370 jobs will be hit in a round of branch closures, TSB said, as it revealed Wandsworth and Twickenham were among the 82 locations set to shut next year.
The bank announced on Monday that it is slimming down its store portfolio.
Chief executive Debbie Crosbie said TSB is facing runaway costs and needs to take difficult decisions. She said: "There is no doubt we have got challenges to face.
Put simply, the biggest barrier for TSB's future is that we're much more expensive to run than our competition."
She stressed the bank has around twice the number of branches per 10,000 customers as many of its rivals.However, Which? money editor Jenny Ross said communities will be hard hit by the branch closures as many still need traditional banking services.
She said: "Our research has found two-thirds of people would find life difficult without convenient access to a branch, while 11 million people lack the confidence to carry out basic banking tasks online.
"If banks want their customers to go online they must ensure they can deliver consistently reliable services - instead of the current rate of regular IT glitches that people endure.
"The next government should urgently intervene with legislation that protects access to cash as a vital back-up and for as long as it is needed."
Which? said nearly 3,400 bank branches have been closed since January 2015, not including the 82 that are set to be lost at TSB. The hardest-hit area is the North West, it said, with the region losing 434 bank branches over the period.
TSB customer banking director Robin Bulloch said: "We realise this is difficult news for our branch partners and will do everything to support those affected to offer voluntary redundancies and redeploy as many people as we can to other roles."
The Wandsworth branch is set to shut in September, while the Twickenham branch will close in November.