TSB says it plans to grow its balance sheet by 40%-50% over the next five years.
The bank is aiming to become a larger player in the current account market, growing from 4.2% to 6% during that time.
Chief executive Paul Pester says it has already seen four to five as many people opening accounts every week since the TSB brand was re-launched last September than it had before.
Growth plans will also see TSB mortgages becoming available through brokers again from the start of next year.
More top news
The government has announced plans to ensure staff receive tips left by customers in bars and restaurants.
Tens of thousands are having do not resuscitate orders imposed without families' consent, according to the Royal College of Physicians.
The prime minister has said the debate over Britain's EU membership has affected his long-term friendship with the mayor of London.