Australian of the Year and domestic violence campaigner Rosie Batty said on Thursday the "unsafe" camps must close.
Nearly 450 staff at Mr Turnbull's former university have also expressed concern about claims of torture by government employees and contractors.
Australia intercepts all irregular boat migrants and holds them in camps in Nauru and Papua New Guinea.
Mr Turnbull has defended the rules, a central policy of former Prime Minister Tony Abbott, by saying they are tough but save lives.
In a letter to Mr Turnbull, Ms Batty said those detained in offshore centres needed care, instead of punishment.
"The centres are, by their very design, unsafe and dangerous places. These centres cannot be patched up," she said.
The academics, meanwhile, said there was "overwhelming" evidence that "repugnant acts" including torture and abuse had been carried out the camps.
This week, the Australian Broadcasting Corp aired allegations made by a Somali woman who said she was raped by two men on Nauru - and that it took police on the Pacific island four hours to respond.
Is Australia's Cambodia solution 'an expensive joke'?
Australia's controversial asylum policy
Mr Turnbull said the rape claims raised in the report were "very alarming", but defended the government's stance on border security.
"The one thing we know is these policies, tough though they are, harsh though they are in many respects, actually do work, they save lives," Mr Turnbull said.
"We're taking a number of steps to work closely with the Nauru government to ensure the safety and the security of all the refugees living in that community."

The 34ft (10m) catamaran crashed into the turbine off the coast of Cumbria, about 2.5-miles (4km) off Walney Island, near Barrow.
An RNLI lifeboat from Barrow rescued the crew and brought them to shore shortly after 02:50 BST.
One of the men, aged 58, suffered a head injury and was taken to hospital for treatment.
The fishing vessel had been travelling from Ravenglass at the time of the incident.
The RNLI towed the vessel back to shore.

Click to enlarge image
A black and white photo of the publication's Manhattan headquarters takes pride of place, with the strapline #lastprintissue.
The nod to Twitter is regarded as a backhanded compliment.
The death of the print edition was caused by falling advertising revenues, as audiences moved online.
From the new year, Newsweek will be a digital-only publication. Editor Tina Brown described it as "a new chapter" for the magazine.
In a defiant editor's letter, she wrote: "This is not a conventional magazine, or a hidebound place.
"It is in that spirit that we're making our latest, momentous change, embracing a digital medium that all our competitors will one day need to embrace with the same fervor.
"We are ahead of the curve."
Ms Brown became editor of the publication two years ago, after it merged with The Daily Beast, a news website she co-founded in 2008.
Newsweek's first edition was published on 17 February, 1933. It made an immediate splash with its front cover, featuring seven photos - one news story for each day of the week.
Although it always took second place to its rival, Time, it  gained prominence in the 1960s for its coverage of the civil rights movement.
At its height, it had a circulation of 3 million, but declining readership and advertising revenue saw it fall into losses.
It was sold by the Washington Post Company to businessman and publisher Sidney Harman for $1 in 2010, and was merged with the Daily Beast three months later.
Ms Brown is a former editor of Vanity Fair and The New Yorker. She unveiled Newsweek's final front cover via Twitter, saying: "Bitter sweet! Wish us luck!"
One reader commented that the hashtag headline was "like using your final breath to ID the killer".
The move to a digital edition will allow Newsweek to cut costs such as printing, postage and distribution. However it will lose money from print advertisers, who traditionally pay more than their online counterparts.
As the final edition went to the printers, The Daily Beast confirmed it would be making many of its editorial staff redundant.
Australia's new PM Malcolm Turnbull is facing fresh calls to shut down offshore migrant detention centres.
[NEXT_CONCEPT]
Three men have been rescued after their fishing boat collided with an offshore wind turbine and began to sink.
[NEXT_CONCEPT]
The 80-year-old US current affairs magazine Newsweek has revealed the image that will grace the cover of its last-ever print edition.