In April, customers were locked out of their accounts for weeks following a botched systems upgrade.
Customers continued to experience IT issues as recently as yesterday.
The bank said the latest problems had been “resolved”, but customers may still experience “a slowness in service”.
In a statement, Pester said: “The last few months have been challenging for everyone at TSB. However, I want to thank all my colleagues across TSB for their dedication and commitment during this period and for their focus on putting things right for TSB customers.”
Pester will be replaced by the bank’s non-executive chairman Richard Meddings with immediate effect.
Meddings said: “On behalf of the TSB Board, I want to thank Paul for everything he has achieved as CEO and pay tribute to the contribution he has made in bringing greater competition to the UK retail banking market.
“Although there is more to do to achieve full stability for customers, the bank’s IT systems and services are much improved since the IT migration.”