"The challenge we are facing at the moment is that while we know everything is working, one of the main ways our customers see everything is working - through our internet banking and mobile app - isn't functioning as well as it should be, and for this I'm truly sorry", he said.
TSB's chief executive Paul Pester has vowed that none of the lender's customers will be left out of pocket as a result of its disastrous IT migration.
The bank will waive all overdraft fees and interest charges for its retail and small business customers for April.
TSB has chose to raise the interest rate on its popular "Classic Plus" account to 5 per cent (a very decent interest rate these days), from 3% - an attempt to prevent customers fleeing to a rival.
Further, for any customer who has faced interest or charges from other providers or lenders as a result of the issues, TSB said it is asking customers to contact it to explain their circumstances so it can work with them on a case-by-case basis to make sure it "puts things right".
The disruption spilled over into branches as people demanded answers and TSB later admitted that several hundred customers had been wrongfully given access to customer data they should not have seen.
When asked if he would resign over the issues, he responded: "I haven't even had time to think about it". On the subject of whether or not he deserved a bonus after the fiasco, he said it would be a matter for the bank.
Asked if he would give up his bonus for this year, he said: "The last thing I'm anxious about is bonuses and pay". I'm focused on putting things right for customers. "It is a decision of the board".
The problem stems from a long-planned switch from the banking platform of former TSB owner Lloyds.
That work was scheduled to be completed on Sunday evening but left customers without access to their money. However, it soon became apparent something had gone badly wrong, with customers unable to login on to accounts or access funds. Others said they were still unable to access the platforms.
Pester said that from 8am on Thursday he was taking over direct responsibility for the IT infrastructure from Sabadell.
"I have drafted in a team of global experts from IBM".
Pester said he was determined TSB would get back on it feet and continue to be a challenger bank to its bigger high street rivals such as Lloyds and HSBC.
Over the weekend and into this week, customers have experienced serious difficulties using TSB services.