"We owe it to survivors not to squander this moment", he said.
The prime minister confirmed the government's intentions in a statement to parliament on Thursday.
He announced the Government would appoint a "survivor-focused reference group" to help in the delivery of an apology to survivors of institutional child sexual abuse.
"It marks a national moment when our country acknowledges that for decades we have failed to afford children one of their most basic rights - to be heard and protected - to be listened to without doubt, to be trusted, to express their views and have them taken seriously".
Mr Turnbull's pledge follows the conclusion of a four-year inquiry that found tens of thousands of children had been abused in Australian institutions.
Mr Turnbull urged the states, territories and non-government organisations not to delay signing up to the scheme, saying only maximum participation would allow it to be successful.More news: North Korea judged victor of diplomatic gold at Olympics
More news: Saudi Scholar says Women Don't Necessarily Have to Wear Abayas
More news: Moon: Team Ash Wednesday or Team Valentine? Pick a side
More than $30 million was promised in the last federal budget to set up the redress scheme, which would pay up to $150,000 to eligible survivors from Commonwealth institutional settings, as well as delivering access to counselling and psychological services.
"Unless the states agree to participate, institutions within their jurisdictions will not be able to join".
He will discuss the scheme with state and territory leaders at the Council of Australian Governments Meeting in Canberra on Friday.
"As of today, not a single dollar has come from any of the states or the institutions whose names and deeds fill the pages of this report", Mr Shorten said.
"If the states and the institutions haven't signed up in the next three weeks I will unload like you've never heard before", Senator Hinch said, adding extra pressure was needed on NSW and Victoria.
Survivor advocate Leonie Sheedy of the Care Leavers Network welcomed the proposed apology but said taxpayer-funded organisations that committed or did not investigate abuse, such as orphanages and the police, should also say sorry.