Keir Starmer has said the online debate about child sexual exploitation is based on lies with politicians “jumping on the bandwagon simply to get attention” as he hit back at Elon Musk.

Responding to questions about social media posts from the billionaire Tesla owner, Starmer said a line had been crossed when the safeguarding minister, Jess Phillips, and others received serious threats as a result of the “poison of the far right”.

The prime minister rejected calls for another review into the Oldham rape gang cases because he said the “utterly sickening” issue did not need any more consultations, it just needed action.

He said a line had been crossed when Musk suggested Phillips deserved “to be in prison” for denying requests for the Home Office to open a public inquiry into child sexual exploitation in the town, and called her a “rape genocide apologist”.

Starmer was launching government plans to tackle NHS waiting lists when he was asked about Musk’s attacks on Phillips, and also the billionaire’s claims that the prime minister should be in prison.

He expressed his concerns about the future direction of the Conservative party under Kemi Badenoch, given the Tories did not condemn Musk’s comments about Phillips.

Speaking on a visit to Epsom hospital in Surrey, Starmer said: “Those that are spreading lies and misinformation as far and as wide as possible, they’re not interested in victims. They’re interested in themselves.

“Those who are cheerleading [the far-right activist] Tommy Robinson are not interested in justice. They’re supporting a man who went to prison for nearly collapsing a grooming case, a gang grooming case. These are people trying to get some kind of vicarious thrill from street violence that people like Tommy Robinson promote.”

Expressing his support for his safeguarding minister, the prime minister said: “And those attacking Jess Phillips, whom I’m proud to call a colleague and a friend, on protecting victims – Jess Phillips has done 1,000 times more than they’ve even dreamt about when it comes to protecting victims of sexual abuse throughout her entire career.”

He added: “We’ve seen this playbook many times, whipping up of intimidation and threats of violence, hoping that the media will amplify it.

“Jess Phillips does not need me or anybody else to speak on her behalf. But when the poison of the far right leads to serious threats to Jess Phillips and others, that in my book [means] a line has been crossed.”

Starmer also defended his record as director of public prosecutions (DPP), which has been called into question in the online debate on child sexual exploitation.

He said: “For many, many years, too many victims have been completely let down; let down by perverse ideas about community relations or by the idea that institutions must be protected above all else. And they’ve not been listened to, and they’ve not been heard.

“And when I was chief prosecutor for five years, I tackled that head-on, because I could see what was happening, and that’s why I reopened cases that have been closed and supposedly finished. I brought the first major prosecution of an Asian grooming gang – in the particular case it was in Rochdale, but it was the first of its kind.”

He added: “We changed, or I changed, the whole prosecution approach, because I wanted to challenge, and did challenge, the myths and stereotypes that were stopping those victims being heard.”

He said his record was “not secret” and that he had called for “mandatory reporting of child sexual abuse”, which he claimed “the Tories did nothing about … including when the [Prof Alexis] Jay report came out”.

Musk had used his own platform, X, to claim Starmer failed to act against the sex offender Jimmy Savile when he was DPP.

Ed Davey, the leader of the Liberal Democrats, said: “People have had enough of Elon Musk interfering with our country’s democracy when he clearly knows nothing about Britain. It’s time to summon the US ambassador to ask why an incoming US official is suggesting the UK government should be overthrown.

“This dangerous and irresponsible rhetoric is further proof that the UK can’t rely on the Trump administration, and it’s in our national interest to rebuild trade and security ties with our allies in Europe.”

John Swinney, Scotland’s first minister, said Musk’s attacks on Phillips were “completely and utterly reprehensible”, and backed Starmer’s criticisms of the US billionaire’s recent social media posts on UK politics.

Speaking to reporters in Edinburgh, Swinney said the best defence against the surge of far-right populism that Musk appeared to be fuelling was for mainstream politicians to repair the UK economy, improve public services and reverse the damage caused by 14 years of austerity.

He said UK political leaders should also review party funding laws to ensure that foreign donors were not able to influence UK elections – a reference to speculation that Musk and other overseas billionaires are planning to fund Reform UK.