Last week's banking scandals demolished a convenient myth: that the banking crash was all the fault of a few colourful rogues like Fred the Shred of RBS and Adam Applegarth of Northern Rock. We have been reminded, instead, that the rot was far more widespread. Incompetence, corruption and greed have been endemic in British banking. The RBS/NatWest computer failure illustrated the incompetence. Millions of households and firms now have to clean up the mess caused by accidental missed payments, bounced cheques and cash shortages.

Computers do crash, but RBS is far too big and it has eloquently shown the folly of over-centralised, mechanised banking systems which have sucked the life out of local, branch-based, relationship banking. We can only gaze with envy at Germany, where highly efficient, locally based, non-profit Sparkassen do what banks are supposed to do: safely channel our savings into productive investment.

The Libor scandal demonstrated deeply corrupt practices at the heart of Barclays Capital, our leading investment bank, and at others too. The public cannot understand how a corporate fine – which will be passed on to customers and shareholders – begins to address the problem.

Then there is the greed: the bonus-driven mis-selling of complex financial derivatives (in the wake of payment protection insurance and other scandals). Tens of thousands of small businesses were told that they could have a bank loan only if they also bought products that neither they nor the bank salesmen understood. Derivatives can be useful hedging investments for companies with sophisticated financial departments but are dangerous when mis-sold to butchers, publicans and fish and chip shops. Many good firms are now being bankrupted as a result. There must be prompt redress and a strict moratorium on enforcing claims.

Faced with a moral quagmire of almost biblical proportions, what should be done? One of the suggestions from Labour is a costly Leveson-style public inquiry. It would certainly be enlivened by Ed Balls explaining why, in government, he allowed the regulatory mess to occur in the first place. Instead we have announced an urgent independent review into the operation of Libor and professional standards for bankers, to enable us to make changes via bills currently going through parliament. And this week we will launch a consultation on criminal sanctions for directors of failed banks, following the recommendations of the FSA report into RBS.

Sorting out the mess involves several steps. First, the banks have to be made safe so that there is no risk of further collapse. This the present government has done by toughening regulation. The problem is that safer banks have become zombie banks, incapable of maintaining a sufficient supply of affordable credit to small businesses and householders to support a growing economy. The government and the Bank of England are together ensuring that banks have capital and liquidity; the challenge now is to ensure that highly paid bankers use them and are rewarded for sensible banking rather than the present combination of hoarding and speculative trading.

The second step is to carry out the government's agreed Vickers reforms, which stop contamination between casino-style investment banking and traditional retail and business banking. Putting investment bankers in charge of high-street banks is like putting mice in charge of cheese supplies. The wisdom of clearly separating investment banking and traditional banking, through ring-fencing, has been reinforced by this week's events and it is now down to parliament to expedite the reforms.

Third, there has to be accountability. Regulators are a backstop: they don't own banks. The governance at the top of our leading banks has been lamentably weak. No one at the top of Barclays will take responsibility for systemic abuse. Shareholders, the owners, have a major responsibility here. I am bringing in legislation to strengthen their control over pay and bonuses, through binding votes, but shareholders have to get a stronger grip on weak boards and out-of-control executives. And our range of criminal sanctions is currently narrow compared with tough jurisdictions like the US.

Last, but not least, cultural change involves different business models. I derive encouragement from the takeover by the Co-op Bank of a chunk of the Lloyds network and the move by Nationwide into business lending. The excellent Swedish business bank Svenska Handelsbanken – which has no staff bonuses – is expanding rapidly. Increasingly, banks are being replaced by non-bank channels. I am a long-term optimist about Britain. Last week's scandals will lead to a better system of banking.