Royal Bank of Scotland nearly doubled pre-tax profits to £2.65 billion in the first half of the year, despite taking an extra £250 million hit for mis-selling financial products.
The state-backed lender published interim results a week early, saying it was because they were "significantly stronger" than market expectations.
Pre-tax profits for the first six months of 2014 were up from £1.37 billion the year before, RBS said.
But for the second quarter they were 38% lower at £1.01 billion as the group took a hit from restructuring costs and provision for compensation for the mis-selling of payment protection insurance (PPI) and interest rate swaps.
Chief executive Ross McEwan said the results showed the underlying strength of the business, but warned there remained "bumps in the road ahead" as it continued to deal with scandals of the past and dispose of toxic assets.
Shares in the lender soared, opening 10% higher.
The bank, which is 80% owned by the Treasury, said the better results came as economic improvements in the UK and Ireland had fed through to its bottom line while it was also running down bad assets more quickly.
Meanwhile in the quarter to the end of June it set aside an additional £150 million to cover PPI mis-selling, taking the total hit from the scandal to date to £3.25 billion.
Its provision for interest rate swaps - complex financial products sold to small firms - rose by £100 million to £1.3 billion.
Meanwhile, the cost of the bank's mammoth restructuring programme came to £385 million for the quarter, and £514 million for the half-year.
RBS had tumbled to a full-year pre-tax loss of £8.2 billion for 2013 as it took billions of pounds of provisions for past scandals and the creation of an internal "bad bank" to hive off toxic assets.
But for the first quarter of this year profits had doubled to £1.64 billion, and results for the second quarter, while lower than last year, were better than expected.
Chief executive Ross McEwan said: "The results we are posting today show the steady progress we are making as we take the steps to be a much simpler, smaller and fairer bank.
"These results show that underneath all the noise and huge restructuring of recent years, RBS is a fundamentally stronger bank that can deliver good results for customers and shareholders.
"There is progress on all of our key priorities - capital is stronger, costs are lower and customer activity is gradually improving - although we have only just started with our programme to make it easier for customers to do more business with us.
"But let me sound a note of caution. We are actively managing down a slate of significant legacy issues. This includes significant conduct and litigation issues that will hit our profits in the months and years to come.
"I'm pleased we've had two good quarters, but no-one should get ahead of themselves here - there are bumps in the road ahead of us."