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The fuel poverty alliance said "woeful" levels of insulation have led to Britain's homes falling "way behind" those of comparable European countries such as Sweden, Germany and the Netherlands.
The alliance said the wholesale cost of gas in the UK was much lower than in most European countries but households paid much higher bills due to the amount of heat lost from homes.
There are more than five million UK households living in fuel poverty, defined as spending more than 10% of their income on energy.
The Energy Bill Revolution said it wanted to see carbon tax spent on an ambitious programme of home insulation, claiming that it could save up to £500 a year on a family energy bill and eliminate fuel poverty in the UK "once and for all".
The groups warned party leaders that focusing on "short-term solutions" to the energy bill crisis, such as price caps, windfall taxes and cutting green subsidies, they were "ignoring the only way to truly solve the energy bill crisis".
Fuel poverty campaigners have written to Prime Minister David Cameron demanding that party leaders act on the "national crisis" of cold homes.
The UK is second only to Estonia among European nations for the number of people who are struggling to pay their energy bills, according to research by the fuel poverty alliance Energy Bill Revolution found.
The alliance, which includes Age UK, Barnardo's, Consumer Futures and National Energy Action, has told leaders that investment in "super insulation" for the nation's homes is the only way to end the "scourge" of fuel poverty and the best way to bring down energy bills.