The new TSB bank has been launched with a promise to "fuel local economies", but the website hit problems on the first morning.
Lloyds Banking Group said there have been "intermittent" website problems across almost all its brands, including the new standalone TSB.
Five million customers are seeing their accounts automatically transferred from Lloyds to TSB.
Customers' bank account numbers and sort codes will remain the same.
TSB's mobile banking will not start until later in the week.
Bank cards will continue to work, with the name of the new bank included as new cards are issued in due course.
Some 631 branches of Lloyds Banking Group, including all of the Cheltenham and Gloucester branches and all Lloyds branches in Scotland, have been switched to the standalone TSB.
The new bank will be sold off next year, as part of a process ordered by the European Commission, to provide greater competition.
The sale was a condition of the government's bail-out of Lloyds, which is 39%-owned by the taxpayer.
Lloyds is currently the UK's dominant personal bank account provider, and will run the TSB as a separate unit until it is sold as a public company in its own right via the issue of shares next year.
TSB's leadership said that the ethos of the bank will mirror the original Trustee Savings Bank with 200 years of history.
"We bring the new TSB Bank back to over 600 communities across the UK to fuel their local economies - and nothing else," said Paul Pester, TSB's chief executive.
"We have today launched a bank which has been born fully formed."
The TSB name is reappearing after TSB merged with Lloyds 18 years ago.
As well as the branches, TSB will have four call centres and employ 8,500 staff, making it the eighth largest High Street bank. In a BBC News interview last week, Lloyds Banking Group's chief executive, Antonio Horta-Osorio, promised that the transition for customers will be "seamless".
However, a number of customers took to social media to complain about internet banking issues on the first morning.
A TSB spokeswoman said some customers were unable to log on successfully, although the problem should now be resolved. Branches, telephone banking and cash machine facilities were not affected in any way.
Lloyds said the internet banking problems were unrelated to the switchover.
The UK's banking industry is dominated by five big names, which control 83% of retail (personal) bank accounts.
As well as Lloyds, the other banks which lead the field are RBS, Barclays, HSBC and Santander.
They have been variously caught up in a catalogue of scandals, including fixing the key Libor interest rate and mis-selling payment protection insurance (PPI).
Later this month, new rules designed to make it easier for people to switch accounts will come into force.
There has been disagreement among commentators over whether the appearance of TSB will boost competition.
Shore Capital's banking analyst Gary Greenwood said: "TSB will be painted as a new challenger brand on the High Street but I doubt that its pricing is going to be very differentiated to competitors.
"Current accounts tend to be very sticky and customers only tend to move if they have a really, really bad experience."
Mark Garnier, a Conservative member of the cross-party parliamentary committee on banks, also voiced doubt: "TSB is certainly not the answer. We want entrepreneurs to come into the marketplace and start opening banks."
However, Kevin Mountford, head of banking at price comparison website Moneysupermarket, said: "The launch of TSB bank can only be good news for consumers as it creates greater competition on the High Street."
Lloyds had been close to selling the business to the Co-operative Bank. That fell through in April after concerns emerged over the Co-op's financial strength.