Jeremy Corbyn has indicated he will not bow to party pressure and move immediately towards demanding a second referendum, after Labour narrowly beat the fledgling Brexit party in the Peterborough byelection.

Corbyn – arriving in the Cambridgeshire city after the party’s candidate Lisa Forbes won by 683 votes, leaving the Tories trailing in third position – called for the “squabbling contenders” within the Conservative party to give the public a general election.

The Labour leader, flanked by Forbes and MP Louise Haigh, who masterminded the byelection victory, told Labour supporters in the city centre the party “is not at the stage yet” to push for a public vote.

Asked about Labour splits over a people’s vote, he said: “Obviously every party discusses its own position and its own strategy. I have said all along that we would put to parliament our proposals on a customs union on a trade relationship and the dynamic protection of consumer and workers’ rights.

“As our conference resolution agreed last September, we would then be prepared to put that to a public vote. We are not at the stage yet where parliament has actually voted on that. I think it would be much better if there was actually a general election.”

Corbyn’s comments came as the internal Labour row about Brexit warmed up after the party’s disastrous result at last month’s European elections.

There has also been unease from some Labour MPs and Jewish campaign groups about Forbes’ election. The new MP had previously apologised for liking a Facebook post that said Theresa May had a “Zionist slave masters agenda”, saying she had meant to “like” the video of children praying in solidarity with the victims of Christchurch attacks, not the accompanying views.

The Times reported on Friday that the Labour MP Louise Ellman was calling for an investigation. She said: “These serious allegations must not be brushed under the carpet. Lisa Forbes should be suspended while the Labour party carries out an investigation.” The paper said the Labour MP Margaret Hodge had also put in an official complaint about Forbes.

The Jewish Labour Movement said Forbes needed to go “far further” to demonstrate to the Jewish community that she was not racist. The Times reported that the group was calling for Forbes’ suspension.

The Labour backbencher Jess Phillips tweeted she could not be as “gleeful or proud as I’d want to be [after the byelection in Peterborough] because of how it shows that antisemitism is becoming normal in the party.”

Phillips added: “Lisa ignored and endorsed antisemitic things, I’ll take her explanation and apology at face value and look forward to her proving, as others have, that actions not excuses alone can heal. But with every case the party’s values chip away and our ability to stand up against hate erodes.”

Haigh, who worked closely with Forbes, said the emergence of the postings was a low point for the campaign. She said: “The whole campaign team were really upset when the posts that Lisa had mistakenly engaged with came to light and that was definitely a very difficult point for the campaign. But Lisa fully accepts they were wrong. She is really, really sorry.

“I will be working with her to engage with the community and deepen her understanding and help her to be an ally in the future.”

On Brexit, grassroots activists are organising for constituency Labour parties (CLPs) to pass motions demanding a second referendum in the run-up to the party’s autumn conference – the same process that laid the groundwork for the shift in policy last year.

Mike Buckley, of the campaign group Labour for a Public Vote, said: “Hundreds of CLPs are already set to debate motions for conference calling for the party to back a new referendum, and to campaign for remain. We’re confident of winning in September – but would much prefer the party to shift now. If we don’t, we’ll lose more voters over the summer and we won’t be able to oppose no deal as effectively.”

The Labour MP Clive Lewis, of the pressure group Love Socialism Hate Brexit, tweeted in response to the Peterborough result: “We’ve blunted our opponent’s momentum with an overwhelming home field advantage. Now we must arm our activists with the radical policies they need to finish the job in the long slog ahead, including a public vote”.

Party insiders said tensions between Corbyn’s office and those pushing for a more full-throated anti-Brexit stance were running high.

The shadow foreign secretary, Emily Thornberry, was dropped as Corbyn’s usual stand-in for prime minister’s questions this week after she spoke out about Labour’s strategy for the European elections. On polling day last month, Thornberry said Labour was “not clear on the one single thing that people wanted to hear”. She was replaced at the dispatch box by Rebecca Long-Bailey, a rising star on the left of the party and a staunch Corbyn loyalist.

Meanwhile, there is growing chatter about the possibility of the party’s rules being changed to create a second deputy leader, alongside Tom Watson, who has pursued a vocal campaign to toughen Corbyn’s Brexit position.

The proposal was mooted last year, but dropped at the last minute because of fears that a vote among party members could be used as a proxy referendum on Labour’s Brexit policy.

A source close to Watson said: “Lashing out by expelling Alastair Campbell, publicly humiliating Emily Thornberry and challenging Tom for the deputy leadership is not the way to resolve this situation.

“No matter how many punishment beatings are meted out, Tom will continue to speak out for what he believes in – a confirmatory ballot and a broad church Labour party that is avowedly pro-European.”

However, one close ally of Corbyn suggested the idea of challenging Watson may just have been “loose talk” in the aftermath of the European elections shock, which was then being “actively briefed by Corbyn’s opponents” to “add to the sense of victimhood” on the right of the party.

In Peterborough, amid cheers from supporters, Corbyn called for Tory leadership contenders to give the general public a chance to vote them out of office. “On the day that Theresa May ceases to be leader of the Conservative party, my message is to all the squabbling contenders for the Tory party leadership: bring it on. We are ready for a general election at any time,” he said.

Forbes won 10,484 votes, beating the Brexit party’s Mike Greene, who took 9,801 votes, a margin of 683. The Conservatives were beaten into third place with 7,243 votes. Turnout was 48%.

Farage left the count through a back door minutes before the result was announced. But Greene said the Brexit party had made a significant breakthrough, despite not winning. “We have shaken up British politics with none of the data that is necessary to win a parliamentary election,” he said. “We’ve had two parties ruling for decades; that’s not happening any more.”