Criticism comes as it emerges that resignation of chief whip Andrew Mitchell was forced by a revolt of young Tory MPs

David Cameron was battling to shore up his authority on two fronts as it emerged that a revolt by young Tory MPs forced the resignation of Andrew Mitchell, and party grandees – led by Lord Tebbit – broke cover to question the government's competence.

With Downing Street still reeling from the decision by Mitchell to quit on Friday, after Cameron spent four weeks defending him, sources said a lack of support from the 2010 intake of Tory MPs finally persuaded him he had insufficient authority left.

News of the "youth revolt" is a serious threat to Cameron's authority, suggesting that dozens of MPs who were overlooked in the recent reshuffle are now prepared to rebel rather than be loyal in the hope of eventual preferment.

Sources said that when Mitchell toured the Commons tea rooms on Wednesday and Thursday, he was taken aback by the opposition to him staying put, despite Cameron's support.

The news comes as the former Tory chairman Norman Tebbit tears into Cameron and the coalition, saying its problems stem not from too many "toffs" but a lack of basic competence.

Writing in the Observer, Tebbit says: "This dog of a coalition government has let itself be given a bad name and now anybody can beat it. It has let itself be called a government of unfeeling toffs. Past governments have had far more real Tory toffs: prime ministers Alec Douglas-Home and Harold Macmillan, or even in Thatcher's day, Whitelaw, Soames, Hailsham, Carrington, Gowrie, Joseph, Avon, Trenchard and plenty more, without incurring similar abuse."

He adds: "The abiding sin of the government is not that some ministers are rich, but that it seems unable to manage its affairs competently."

Mitchell quit over claims that he called police officers "plebs" and "morons" – something he strongly denies. He has admitted that he swore at police when they barred him from taking his bike through the main Downing Street gate.

The resignation, coupled with claims that the chancellor, George Osborne, sat in a first-class train seat with a standard-class ticket, followed a week of confusion over energy policy. The events earned the Tory party fiercely critical headlines on Saturday, even in normally supportive papers such as the Daily Mail.

The Observer understands that Tebbit's concerns are shared by a growing number of senior Tory donors who believed Mitchell should have been sacked earlier. Sources said some of the largest contributors to the party are also questioning whether Cameron has the clarity of purpose to lead the coalition.

In a scathing critique Tebbit says Cameron needs to find leadership skills and impose "some managerial discipline not just on his colleagues but on himself. Had Mr Miliband concentrated his fire on a long list of muddles from the proposed sale of our national forests to the BAE and energy policy muddles of recent days it would have been far worse."

A new Opinium/Observer poll, conducted before the Mitchell resignation and Osborne rail fare furore, puts Labour on 40%, nine points ahead of the Tories on 31%. The Lib Dems are on 9%, just behind Ukip on 10%. Despite Labour's lead, the Tories still have a slight edge on the economy. When asked who they would trust most to run the economy, 30% said the Tories and 29% Labour.

On Saturday Downing Street tried to hit back, insisting it was time to trumpet the government's achievements in cutting the deficit, unemployment and NHS waiting lists and increasing the number of people in work. On Monday Cameron will make a major speech on crime. His new "tough but intelligent" approach will signal the end of his infamous "hug a hoodie" image on law and order.

Tory MPs and ministers said they were becoming exasperated about policy failings, particularly after the PM appeared to announce a new policy on energy prices of which the Department of Energy and Climate Change had been unaware.

The Labour leader, Ed Miliband, was on Saturday among dozens of speakers at a TUC-organised march in London against austerity. Other rallies took place in Glasgow and Belfast.

Miliband said: "Andrew Mitchell may finally have resigned. But the culture of two nations runs right across this government. They cut taxes for millionaires. And they raise taxes for ordinary families. They leave young people out of work while the bonuses at the banks carry on. They even have a chancellor of the exchequer who tries to travel first-class on a standard-class ticket. It's one rule for those at the top and another rule for everybody else: everybody like you who plays their part and does the right thing. The trouble with this government is that while they are think they are born to rule, it turns out they are not very good at it."

Grant Shapps, the Tory chairman, said Miliband was leading Labour back "to the days of Michael Foot".

"The world has changed. Britain is in a global race. That's why we're dealing with our debts, getting people back to work and keeping mortgage rates low.

"Ed Miliband and his union paymasters are turning a blind eye to all that. By marching with them, Ed Miliband proved he still stands for more spending, more borrowing and more debt.