He said he knew this was what the leaders of Westpac expected.
Turnbull said that during the global financial crisis - "or what probably should be better called the global banking crisis" - the Australian public, through the government, had provided the banks with vital support.
"Australians understood that we needed to ensure our banks kept trading, that a strong well-regulated financial sector in Australia was a great blessing, a great national asset," he said.
He said today, many Australians were asking: "have our bankers done enough in return for this support?
"Have they lived up to the standards we expect, not just the laws we enact?"
Wise bankers recognised these were legitimate questions, Turnbull said. "Dismissing them as bank bashing misses the point."
"The truth is that despite the public support offered at their time of need, our bankers have not always treated their customers as they should.
We expect our bankers to have higher standards, we expect them always, rigorously, to put their customers' interests first.
"Some, regrettably as we know, have taken advantage of fellow Australians and the savings they have spent a lifetime accumulating, seeking only dignity and independence in their retirement."
Turnbull said that redressing wrongs was important, especially when "done promptly and generously".
"Wise bankers understand that banks need to very publicly demonstrate that their values of trust, integrity, placing the customer first in every way - these must be lived and not just spoken.
"They recognise that remuneration and promotion cannot any longer be based solely on direct financial contribution to the bottom line. Employees who live those values, impart them to others and call out those who do not should be rewarded and recognised and promoted in a healthy banking culture.
"The singular pursuit of an extra dollar of profit at the expense of those values is not simply wrong but it places at risk the whole social licence, the good name and reputation upon which great institutions depend.
"Now all business is about more than just a profit or a new product - it's about building opportunities for Australians, customers and staff and making a greater contribution to our nation."
Nationals senator John Williams renewed his call for a royal commission into the financial sector. "First cab off the rank was Storm Financial. Then we went through the liquidators industry, where there's some very bad eggs.
"Then of course we had the financial planning scandal. Now we've just had the managed investment schemes where billions of dollars were lost. Now the life insurance industry - and of course these latest allegations of bank bill swap rates.
"As time goes by the case builds stronger and stronger, in my opinion, for a royal commission into the finance sector. My concern is the culture is simply profit, profit, profit and to hell with the customers."
Speaking to journalists after his address, Turnbull dodged a question on whether he would support a Royal Commission.
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