Jeremy Corbyn will use a speech in London to urge Labour voters to lay the blame for pressures on housing, jobs and the NHS at the door of the Conservatives, instead of seizing on the EU as a scapegoat.

As the referendum campaign steps up a gear, with Britain Stronger in Europe saying it plans to hold 1,000 events over the weekend, the Labour leader, a long-time sceptic when it comes to the EU, will tell his audience at a rally on Saturday that voting to leave would not help tackle the UK’s problems.

“People in this country face many problems, from insecure jobs, low pay and unaffordable housing to stagnating living standards, environmental degradation, and the responsibility for them lies in 10 Downing Street, not in Brussels,” he will say.

“This country is being let down by a Conservative government that is failing on housing, failing our children, failing our NHS, failing to create good quality secure jobs and consistently failing even to meet its own economic targets.”

The main party leaders will be out pushing for a remain vote this weekend, with David Cameron unveiling a new campaign poster and the Liberal Democrat leader, Tim Farron, meeting voters in Kendal.

Corbyn’s deputy, Tom Watson, will take the Labour in for Britain battlebus through the West Midlands, visiting Telford, Wolverhampton and Birmingham.

Cameron said: “This is a day unlike any other: politicians of every stripe taking to the streets with the same message. Because we face a vote unlike any other, one which will shape our country for decades – even generations – to come.”

Robert Oxley, of Vote Leave, said the remain camp were “playing catch-up”. He claimed he had 20,000 activists signed up to get involved, and said Cameron, whose party is deeply split on the issue of Europe, was relying on Labour ground troops to win the referendum.

“Most of the in campaign is based around Labour, but being told by No 10 what to do,” he said.

Vote Leave has said David Cameron is relying on Labour ground troops to win the referendum. Photograph: WPA Pool/Getty Images

Labour voters, particularly younger ones, are regarded as crucial to achieving a remain vote on 23 June, and senior figures in Stronger In are concerned that Corbyn has been too reticent about making the case. However, his team are anxious about the possibility of being seen to ally themselves too closely with the Conservatives.

The collapse of Labour’s support in Scotland followed its decision to mount a joint push against independence with the Conservatives, led by Alistair Darling, in the 2014 referendum. Labour was pushed into third place in the Holyrood parliament, behind the Conservatives, in last week’s elections.

With polling out of the way in Scottish, Welsh and local elections, senior Labour figures have switched their focus to the referendum. The former party leaders Gordon Brown and Ed Miliband have made speeches, and Tony Blair is expected to make his own intervention in June.

Corbyn will say that a Labour government would use Britain’s EU membership to press for different priorities across the continent, including improving workers’ rights.

“When Labour comes into government we will work with our allies across the continent to reform the European Union to increase democratic accountability to strengthen workers’ rights and the scope for public enterprise ... and to work together to tackle issues like tax avoidance and climate change,” he will say.

Christine Lagarde, the managing director of the International Monetary Fund, became the latest policymaker to issue a warning about the consequences of a Brexit on Friday, saying they could range from “pretty bad to very, very bad”. Mark Carney, the governor of the Bank of England, said a vote to leave could trigger a recession.