Boris Johnson has promised to present the delayed online harms bill to parliament before Christmas in the wake of the killing of Sir David Amess, after Keir Starmer urged him to work collaboratively in clamping down on web-based extremism.

The prime minister also appeared to agree to another of Starmer’s requests, that the bill would include a commitment to possible criminal sanctions against tech company bosses who do not do enough to remove harmful or illegal content.

Pressed by the Labour leader at prime minister’s questions on what measures he would take, Johnson agreed with the idea of a cross-party approach, but also peppered his responses with partisan criticisms, prompting some shouts of complaint in the Commons.

Starmer began his questions by noting the tributes to Amess, the veteran Conservative backbencher who was killed at a constituency surgery on Friday, given by MPs from all parties in the Commons on Monday.

“I want to see if we can use that collaborative spirit to make progress on one of the issues that was raised on Monday – tackling violent extremism,” Starmer said, asking about progress for the online safety bill, first mooted three years ago.

Starmer said: “Will the prime minister build on the desire shown by this house on Monday to get things done, and commit to bringing forward the second reading of the online safety bill by the end of this calendar year? If he does, we’ll support it.”

Johnson replied by initially pledging the bill would complete all its stages before Christmas, before correcting himself – seemingly at the prompting of the home secretary, Priti Patel, who was sitting next to him – to say he would “bring it forward before Christmas in the way that he suggests – and I’m delighted that he is offering his support”.

Starmer then pressed on potential changes to the proposed bill, saying: “Tough sanctions are clearly needed, yet under the government’s current proposals, directors of platforms failing to crack down on extremism would still not face criminal sanctions. Why is that?”

In a later answer, Johnson appeared to agree to this, saying: “And yes, of course we will have criminal sanctions with tough sentences for those who are responsible for allowing this foul content to permeate the internet.”

While Johnson initially praised Starmer “for the spirit in which he has approached this issue”, the PM appeared to bristle at the Labour leader’s suggestions on how to deal with the problem, and his collegiate spirit appeared to lapse.

Several times Johnson criticised Labour for voting against the new police, crime, sentencing and courts bill. Labour did this because of concerns at elements such as greater police powers to crack down on protests, but Johnson has repeatedly and incorrectly said it was because of objections to tougher sentencing rules in the bill.

“I’m all in favour of a collegiate and cooperative approach,” Johnson said. “In which case I think it would be a fine thing if the opposition would withdraw their opposition to our measures to stop the early release of serious extremists and violent offenders, that’s all I’m try to say.”

Starmer said: “Really. After the week we’ve just had, I really don’t want to descend to that kind of knockabout. Either we take this seriously and go forward together – and I’m taking my lead from those on the opposite benches on Monday, and what they were saying about the need to tackle this – or we do a disservice to those that we pay tribute to.”