Jeremy Corbyn has said he is pleased Labour has "hung on" to councils in England and told his internal critics to remember he was elected party leader with a "large majority".
David Cameron said the results of Thursdays elections, which saw Labour pushed into third place in Scotland, showed there was a "realignment" taking place in British politics.
Several Labour MPs have warned Corbyn the results are not good enough, with one claiming the "clock is ticking" on his leadership.
However speaking in Sheffield, Corbyn said: "All across England last night, we were getting predictions that Labour was going to lose councils, we didn’t, we hung on and we grew support in a lot of places."
And he said he wanted to "invite" his critics to "join" him in opposing the Conservative government. "I was elected a year ago with a very large majority and a very large mandate," he added.
At the time he was speaking, Labour had lost 22 council seats in England. But the party managed to retain councils in the south including Crawley and Southampton.
Cameron said Labour was "so obsessed with leftwing causes" it forgot what voters really wanted.
"Local election day for sitting prime ministers is meant to be a day of dread, it's meant to be a day when you are sitting there waiting for someone to knock on the door like the condemned man waiting for hangman, but that wasn’t what it was like last night," he said.
Speaking in Peterborough, the prime minister said: "The Labour Party has completely lost touch with the hardworking people they are supposed to represent."
Labour's deputy leader Tom Watson has said it would be “unfair” for Labour MPs to blame Corbyn for the party’s performance.
“After eight months it would be very unfair and improper actually to hang this set of election results on Jeremy Corbyn’s peg alone,” Labour’s deputy leader said. "After only after eight months people would say he needs more time to show that direction that Labour needs to take to win that 2020 general."