The UK has spent almost £2bn housing vulnerable homeless families in short-term temporary accommodation, according to figures that demonstrate the scale of Britain's housing crisis.

Rising private rents, a shortage of affordable housing and benefit cuts have forced local authorities, particularly in London, to place increasing numbers of households in bed and breakfast accommodation, hostels and shelters.

With the number of houses built in Britain falling to new lows, according to figures released last week, a four-month study by the Bureau of Investigative Journalism, has revealed that £1.88bn has been spent on renting temporary accommodation in 12 of Britain's biggest cities over the past four years.

Campaigners have said welfare changes will exacerbate the problem. Official figures show that in London alone 7,000 families dependent on benefits stand to lose more than £100 a week under the benefit cap, and many are expected to become homeless as a result.

Leslie Morphy, chief executive of the homelessness charity Crisis, said: "For the sake of cutting just a few pounds a week from their benefits, families and individuals are being forced out of their homes, to be put up in B&Bs or temporary accommodation that costs us all far more."

A separate investigation by the Bureau of Investigative Journalism has uncovered evidence that London councils are rapidly accelerating the rehousing of homeless households outside their home boroughs. Some 32,643 homeless households have been rehoused out of their borough since 2009.

In the year to April, 10,832 households were rehoused in this way – a 16% rise on the previous 12 months. Most left the more affluent districts of inner London for the cheaper outer suburbs, although an increasing number of London's homeless are being moved to towns outside the capital, such as Dartford in Kent, Slough in Berkshire and Spelthorne in Surrey.

The "destination" boroughs have said the influx of households has put a significant strain on local services. Councillors in Enfield in outer London, where more properties and B&B rooms are secured by London authorities than anywhere else, have said the demand from inner London authorities is pushing up private rents and placing untenable pressure on school places.

"The pressure will not abate," said Edward Smith, a Conservative councillor in Enfield. "Before long we will have to build more secondary schools."

The Labour leader of Slough council, Robert Anderson, said: "If authorities put people in our area with complex needs, or even just families; they need to inform us. If we know where they have come from we can make sure the borough does not shirk responsibilities and just pass on their more difficult clients. You can't just pitch up halfway through a year and expect to get a school place. It's not McDonald's."

The housing minister, Mark Prisk, insisted on Sunday night that councils should be careful about placing families in B&Bs far from their home borough. "There is absolutely no excuse for families to be sent miles away without proper regard for their circumstances, or to be placed in unsuitable bed and breakfast accommodation for long periods of time," he said. "The law is clear: councils have a responsibility to take into account people's jobs and schools when securing homes for those in need."

But Prisk also defended the policy of removing families on benefit from central London. "Nor is it right that those living on benefits should be able to live in parts of the capital that those who aren't reliant on this support couldn't afford to," he said.

Households accepted as homeless by their local council will often be placed in temporary accommodation until a more permanent home can be found for them.

As latest government figures show there were 53,130 households living in temporary accommodation at the end of 2012 – 9% higher than the previous year – a leading law firm is preparing a class action against councils that keep families in B&B for longer than the statutory maximum of six weeks. It is believed a third of British local authorities are in breach of the limit, largely because of a shortage of suitable temporary accommodation.

Official guidance says B&B accommodation should be avoided wherever possible. Lack of privacy and amenities for cooking and laundry means it is "not suitable" for families with children or pregnant women "unless there is no alternative accommodation available and then only for a maximum of six weeks".

Bureau data shows the amount spent on temporary accommodation across 12 of Britain's biggest cities was up 5.7% to £464m last year. And London councils have budgeted for further significant overall rises this financial year.

Since 2009, London councils have secured 5,827 properties and B&B rooms in the three London boroughs of Enfield, Waltham Forest and Haringey alone.

The borough suffering the worst homelessness crisis in the country appears to be Newham, in east London which has spent £185.2m placing people in temporary accommodation since 2009.