The Independent Inquiry Into Child Sexual Abuse has delayed hearings into the late Lord Janner.
An inquiry official said there was "significant overlap" between its investigation and criminal proceedings, which could cause prejudice in court.
Lord Janner's family are calling for the "deeply flawed" inquiry into him to be dropped.
Labour said the whole abuse inquiry was facing a "crisis of credibility", but the PM said she had confidence in it.
Lord Janner died before a court could examine 22 allegations against him of sex offences against boys between the 1960s and 1980s.
Leicestershire Police are still investigating other men, who have also been accused by those who made allegations against Lord Janner.
The Independent Police Complaints Commission is also investigating the way the Janner case was dealt with by police.
The IICSA had been due to start public hearings on 7 March but officials say these need to be delayed because they might involve alleged victims of Lord Janner who also have to give evidence in other investigations.
The IICSA said in a statement that it wanted to "avoid potential issues around witness overlap" with police and the IPCC, but was still committed to holding oral hearings.
Lord Janner's son, Daniel Janner QC, said the delayed hearings were part of a "deeply flawed and disgracefully unfair strand" of the inquiry and they should be dropped.
"The correct place for justice is a court of law and not the inquiry, where proper cross-examination of false allegations is not allowed," he added.
"My father has been silenced through death and so any pretence at fairness or justice is a farce."
'Clearly not working'
The IICSA was set up to examine whether public bodies, including the police, had failed in their duty to protect children from sexual abuse and to examine allegations of abuse involving "well-known people".
The inquiry has suffered a series of setbacks, including resignations of three chairwomen and senior lawyers.
On Wednesday, another of the lawyers involved, Aileen McColgan, quit amid concerns about the inquiry's leadership.
It is now on its fourth chairwoman after Professor Alexis Jay, who led the Rotherham child abuse inquiry, replaced New Zealand judge Dame Lowell Goddard in August.
Shadow home secretary Diane Abbott said Prime Minister Theresa May and Home Secretary Amber Rudd needed to explain to MPs what was going wrong.
"The inquiry was established by Theresa May and it clearly isn't working," she said.
"The victims of child sex abuse have put their faith in this inquiry. The very least they deserve is some openness from government and an explanation for what is going on."
Earlier, Labour's Yvette Cooper said the home affairs select committee, which she chairs, had contacted the lawyers who had left and Lowell Goddard to learn what had gone wrong with the investigation.
"What's unclear is how much this has been about legacy problems - and clearly some personality issues - and whether there have also been structural issues over the last two years, or whether there are ongoing problems for the future," she told BBC Radio 4's Today programme.
The inquiry's third chairman, Lowell Goddard, resigned in August. She has instructed lawyers to fight newspaper claims that she had made racist remarks.
Professor Jay has also been under criticism from survivors' groups because of her background in social work - a profession they say has failed them.
At Prime Minister's Questions on Wednesday, Mrs May said it was important the inquiry continued and that she had "full confidence" in Prof Jay and her ability to lead it.
Meanwhile, the Independent Police Complaints Commission has confirmed it is investigating the conduct of five officers linked to the Met's Operation Midland investigation into paedophile allegations against VIPs - which closed in March without a single arrest or charge.
They range from a deputy assistant commissioner to a detective sergeant.
One of them will also be investigated over his involvement in Operation Vicente - which looked into allegations made about the late Sir Leon Brittan.
This IPCC investigation is separate from the one looking into the handling of the Lord Janner case.
An independent review of Operation Midland by Sir Richard Henriques, released last week, found "numerous errors".
Operation Midland was launched after claims that boys had been sexually abused by a group of powerful men from politics, the military and law enforcement agencies at locations across southern England and in London in the 1970s and 1980s.
It relied heavily on evidence from a complainant known by the pseudonym "Nick", who is now being investigated by the Met for allegedly attempting to pervert the course of justice.