The bank had advised customers that internet banking and services such as money transfers would not be available over the weekend, as it worked on its IT systems.
However, by Sunday evening the service was not fully restored and customers were still reporting problems on Monday.
People complained of not being able to access their accounts and, in some reported cases, customers were shown incorrect account details.
Many customers took to social media to vent worries about data protection and anger about not being able to carry out essential transactions such as making mortgage payments.
A TSB spokesperson said: “We are currently experiencing large volumes of customers accessing our mobile app and internet banking which is leading to some intermittent issues with people accessing our services.
“We are really sorry for the inconvenience this is causing our customers and want them to know we are working as hard and as fast as we can to resolve this problem.”
Customers were still complaining of issues on Monday afternoon.
One Twitter user Danny Parekh said: “All I need is a copy of statement for a mortgage application @TSB yet I can’t log on, your branch wasn’t working properly meaning I spent my lunch in a non-moving queue and now your phone lines aren’t even being answered?! Sort it out #tsb #tsbdown #leaving #useless”
Another Charles Waite wrote: “I have a mortgage to pay and some building work that needs paying for.
“I’m worried that somebody may have access [to] my personal information.”
Fabi-Rad said: “My husband TSB customer very angry our mortgage is due to be paid and you’re holding our money without our consent.”
Iain de’Gresley wrote: “COME ON @TSB !!! I’ve got a mortgage to pay, need to transfer funds to Paypal and have no idea if my wages from last Friday have gone in !!! GET IT SORTED!!”
All I need is a copy of statement for a mortgage application @TSB yet I can't log on, your branch wasn't working properly meaning I spent my lunch inn a non moving queue and now you're phone lines aren't even being answered?! Sort it out #tsb #tsbdown #leaving #useless
— Danny Parekh (@DannyParekh) April 23, 2018
— Iain de'Gresley (@DoctorKiln) April 23, 2018
Something tells me we will end up forking out for late payment and missed mortgage payments #tsbdown
— Cristina Ormesher (@cristina_o9) April 23, 2018