TSB chief executive Paul Pester will give up his "integration bonus" of £2m following the IT fiasco at the bank.
However, his full pay and bonus package will not be decided until the end of 2018, the Treasury Committee has heard.
He told the committee that the bank had received 40,000 complaints after millions of customer accounts were moved to a new computer system.
MPs heard that no date could be set for the end of the fiasco, which has already lasted nearly two weeks.
"If there is one decision in my life that I could change it was the decision to go ahead with the migration. Clearly that was a terrible decision for our bank, for our customers and for me personally," Mr Pester said.
Mr Pester argued that the migration of billions of customer records was successful "to the penny".
The underlying engine of the bank was working well, he said, and most customers could log in normally, although there was an immediate angry reaction from those customers who could not make payments or get into the system.
He said that he could be trusted to carry on the work of the bank, and MPs heard that he would not be quitting.
"They should trust me because I will ensure I will bring TSB out of the problems we're in. I have promised customers they will not be out of pocket," he said.
"[The buck] stops with me. Of course I take absolute responsibility for what has happened to TSB customers."
Mr Pester gave evidence alongside TSB chairman Richard Meddings, and Miquel Montes, chief operating officer at TSB's Spanish owner, Sabadell, who also sits on the TSB board.
During evidence, the TSB bosses said:
- The IT work was not rushed through owing to financial incentives
- Tests before the switch were "misleading" as they did not foresee the problems with the system's capacity
- A law firm has been called in to carry out an independent review of the saga
- Waiting times on phones was "very, very poor"
Mr Pester said that the "accessibility" problems for customers would "not be simple to fix". He admitted that only just over half of complaints had been "acknowledged".
Nicky Morgan, who chairs the committee, said: "What we are hearing is the most staggering example of a chief executive who seems unwilling to realise the scale of the problems that are being faced."
'Stressed and disappointed'
The timing of the TSB disruption has been particularly bad for Lorna McHale, who will be marrying Ben Connolly on Saturday.
The couple have their savings for the wedding in a TSB account which they have been unable to access.
"We had no idea how much money was in there as we got bits and bobs for the wedding," said Ms McHale.
The 26-year-old had to spend the morning before a dress fitting in a branch, which also had IT issues, but did not like the idea of taking out a large sum in cash.
She said she had to "ring to grovel" with suppliers for their wedding, including the DJ, the wedding car provider, and those doing her hair and make-up, all of which were small businesses.
She still cannot access online banking, as the bank's records are now using an old mobile number to which they sent the access code. The bank phoned her when she was in a meeting, and when she called back, the wait was going to be so long that she got cut off.
"I am stressed and disappointed. It is one thing that I really did not need right now," said the bride-to-be.
"I want to know what they are going to do to prevent it from happening again, and how they are going to compensate people for the undue stress."
Her case was raised during the Treasury Committee hearing and TSB chairman Mr Meddings said she would receive full compensation, which would include redress for stress. The compensation would be devised by accountancy firm Deloitte.
What the IT issue is all about
TSB shut down services for two days from the evening of Friday 20 April to move customer data from former owner Lloyds to a new IT system managed by Sabadell.
As soon as the new system was switched on during Sunday evening, customers reported seeing other people's account details alongside a range of other difficulties.
Access to accounts via online banking and the bank's app was patchy in the following days, while those who did manage to get into their accounts encountered some extraordinary errors.
In a letter to MPs, Mr Pester also revealed the extent of "widespread" failures of technology in the branch network. Problems were still experienced in 40% of branches by last Friday.
Frustrated customers also faced average waits of up to an hour for those telephoning the bank.
Last Thursday, experts from computing giant IBM were called in to try to help solve the crisis.