Labour has gained several councils from the Tories, who have suffered the loss of hundreds of council seats across the country in election results so far.
Labour leader Sir Keir Starmer said his party was "back on track" to succeed at the next general election.
Boris Johnson conceded the Tories had had a "tough" time in some areas - but argued results had been mixed overall.
The BBC projects Labour would take 35% of votes and the Tories 30%, had all parts of Britain gone to the polls.
That five point lead gives Labour its biggest local election lead in a decade.
The findings - based on the results declared so far from areas which voted on Thursday - put the Liberal Democrats on 19% and other parties on 16%.
The Liberal Democrats have enjoyed good results, gaining councillors and putting in an improved performance in some Tory heartlands.
With around three-quarters of councils across Britain having declared, results so far include:
The full English results - as well as those for councils in Scotland and Wales and for the Northern Ireland Assembly - will be announced later on Friday and on Saturday.
Ahead of the elections, the Conservatives had been braced for big losses in England, as they faced attacks over the rising cost of living and the ongoing Partygate scandal, including the prime minister's fine for breaking lockdown rules.
Speaking on a visit to Ruislip, north-west London, Mr Johnson said: "It is mid-term. It's certainly a mixed set of results.
"We had a tough night in some parts of the country, but on the other hand, in other parts of the country you are still seeing Conservatives going forward and making quite remarkable gains in places that haven't voted Conservative for a long time, if ever."
He added that he took personal responsibility for the results, which showed the need for his party to renew its focus on helping people with rising living costs.
Although Conservative losses are not as bad as some in the party had warned, there was still criticism of Mr Johnson from his own side.
The Conservative leader of Carlisle City Council, John Mallinson, said the prime minister "bears a lot of the responsibility" for the results and was a "poor option" to lead the party into the next general election.
Ravi Govindia, leader of the Wandsworth Conservatives, said: "Let's not be coy about it. Of course national issues were part of the dilemma people were facing."
And Conservative MP David Simmonds said Mr Johnson had "difficult questions" to answer, adding: "Overwhelmingly the message that I heard on the doorsteps was people were broadly positive about the government's policies, but they are not happy about what they have been hearing about Partygate."