The UK's financial regulator said the failings were "widespread and serious" and led to "significant disruption."
The problems began in April 2018 when an attempt to move data to a new computer system went wrong.
All TSB's services were affected, including branch, telephone, and online banking.
The problems were not fully resolved for eight months, and "a significant proportion of its 5.2 million customers were affected by the initial issues", the Financial Conduct Authority (FCA) said.
Mark Steward, executive director of enforcement and market oversight at the FCA, said: "The failings, in this case, were widespread and serious, which had a real impact on the day-to-day lives of a significant proportion of TSB's customers, including those who were vulnerable."
The FCA said TSB had paid £32.7m in compensation to customers affected by the problems.
TSB chief executive Robin Bulloch said: "We'd like to apologize again to TSB customers impacted by issues following the technology migration in 2018. We worked hard to put things right for customers then and have since transformed our business."
The problems were triggered when TSB tried to move 1.3 billion customer records from an old system run by its former parent bank, Lloyds, to one managed by its current Spanish owner, Sabadell.
However, it proved disastrous, with many customers being locked out of their accounts and some customers being given access to the confidential records of others.
Cases included people trying to complete house purchases and unsure whether they could move in. The BBC spoke to one couple who sat with a loaded removal van outside their new home, waiting for funds to be released.
Another couple locked out of their wedding savings were forced to grovel to their DJ and wedding car provider because they could not pay them as the big day approached.
Businesses were also left in difficulty. Lee MacDonald, who played Zammo in the children's TV show Grange Hill in the 1980s, told BBC Radio 5 live that he was "having an absolute nightmare" in his locksmith business.
Shortly after the issues emerged, Mr. Pester admitted to the BBC that the bank was "on our knees."
The problems continued for many weeks, and TSB faced fierce criticism for its IT failings. MPs directly and unusually criticized Mr. Pester for the saga.
The FCA said while the data itself was transferred successfully, the IT platform "immediately experienced technical failures."
"This resulted in significant disruption to the continuity of TSB's banking services, including branch, telephone, online, and mobile banking," it said.
Mr. Steward added: "The firm failed to plan for the IT migration properly, the project's governance was insufficiently robust, and the firm failed to take reasonable care to organize and control its affairs responsibly and effectively, with adequate risk management systems."
TSB was fined £48.65m - a £29.75m penalty from the FCA and a fine of £18.9m from the Prudential Regulation Authority (PRA).
"The disruption to continuity of service experienced by TSB during its IT migration fell below the standard we expect banks to meet," said Sam Woods, the PRA's chief executive.
In February 2019, TSB said that the disastrous IT upgrade cost £330m, and about 80,000 customers had switched their accounts away from the bank in 2018.
The complexity of the switchover and the problems it caused have meant a lengthy investigation and eventual fine by the regulator. In the meantime, there has been significant reputational damage for the bank.
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