TSB boss Paul Pester has told the BBC the bank is "on our knees" after six days of computer chaos, but pledged "we will get up and come back fighting".
The bank has been forced to call in outside help to help fix the banking system breakdown.
Mr Pester, said he would take direct control of the issue, and had drafted in experts from IBM, who would report "directly" to him.
He also promised no customer would pay overdraft fees or charges for April.
The problems arose last Friday when the bank attempted to move customers' accounts from former owner Lloyds to a platform, newly built, by its Spanish-owner Sabadell.
But since the migration began, around half of TSB's customers have been unable to access their online accounts.
Asked by the BBC if he would give up his bonus for this year, Mr Pester said: "The last thing I'm worried about is bonuses and pay."
When asked if he would resign over the issues, he responded: "I haven't even had time to think about it".
A TSB manager's tale of woe
The BBC has spoken to a fed-up TSB branch manager who made it clear that it's not just customers who are suffering.
"Multiple staff members have been sent home physically and emotionally exhausted."
"In my branch, every member of staff has been in tears and all saying this has been the worst working experience of their lives."
"It is heartbreaking for me as a manager seeing my amazing team break down one by one."
"I am desperate for chief executive Paul Pester to recognise how awful it is in the branches."
The BBC asked Mr Pester what sort of expenses TSB customers could be compensated for. He was told about Phil, a tiler who has lost £2,000 worth of business because he could not pay contractors.
Do people like him have a case for compensation? "Of course they do," said Mr Pester. "We will work through what we think and he thinks is reasonable.
He added: "TSB has caused this problem, we should be able to provide a service to our customers, and we haven't.
"The last thing we are going to be doing is penny-pinching when we are dealing with an issue with our customers."
However, despite Mr Pester's attempts at reassurance, exasperated TSB customers have continued to vent their fury on Twitter.
No they aren't, I still can't make transfers and payments via Internet Banking as your 'state of the art' (your words not mine) upgrade keeps telling me that my password is 'too short and too long' 🤬— John Donnelly (@LeyburnSols) April 26, 2018
Your internet banking is still not working !! Day 5 that i have been unable to log in to my business account. This is a farce !!— derek coventry (@silversports) April 26, 2018
Incredible that after almost a week Im still unable to login to my @tsb account. The standardised response throughout the week to customers who are all deeply concerned about their finances has been nothing short of embarrassing. Shame on you @PaulPester— David Myring (@davidmyring) April 26, 2018