The latest snapshot on the progress to reduce Indigenous disadvantage reveals three of the seven targets are on track to being met, compared with just one last year.
"I can advise that three of the seven Closing the Gaps targets are on track this year, giving us the most promising results since 2011," Mr Turnbull said on Monday, as he delivered the tenth Closing the Gap report in Parliament.
The government says it's halving the gap in child mortality rates and Year 12 attainment, as well as enrolling more Indigenous children into school.
"14,700 Aboriginal and Torres Strait children are enrolled in early childhood education the year before full-time school. We must continue to ensure those children are attending, not just enrolled, in order to receive the full benefit," he said.
But the four remaining targets, which include literacy and numeracy, employment, school attendance, and a key measure to close the 10-year life expectancy gap between Indigenous and non-Indigenous Australians by 2031, are lagging.
"That has to change," Mr Turnbull said. "The last decade has given us a richer understanding of what's working and what's not," he added.
"When it comes to closing the gap we cannot be content with aiming for anything less than proper equality."
Mr Turnbull says the solution to Closing the Gap "rests within the imagination, ingenuity, passion and drive of Indigenous people themselves".
"Governments must be the enabler of this success. Too often we are quick to highlight the despondency which does nothing to help those who aspire," he said.
"The Government will do all that we can to ensure that Indigenous Australians play a central role in our nation's success story in every respect."
Mr Turnbull also says jobs and starting up businesses are vital to improving the lives of First Australians.
"We cannot Close the Gap if we do not have equal participation in the economy."
New Indigenous Grants policy announced
The government unveiled a new Indigenous Grants Policy, which aims to increase the number of Indigenous owned and controlled organisations involved in service delivery.
"Organisations will need to demonstrate value for money and that they have the skills and expertise to deliver the grant," the Indigenous Affairs Minister Nigel Scullion said in a statement.
"Where non-Indigenous organisations are providing services, they will be required to employ local Aboriginal or Torres Strait Islander people and include local Indigenous businesses in their supply chains."
Closing the Gap refresh
On Friday, the Council of Australian Governments (COAG) met with Indigenous leaders to discuss a refresh to the Closing the Gap strategy - the first time Aboriginal and Torres Strait Islander people addressed COAG.
They urged governments to take the drastic measures needed to close the gap. The Prime Minister said a new strategy would be focused on a strengths-based approach on existing targets but also include new targets that address health and education.
"We've agreed to extend the timeframe for concluding this refresh consultation to October this year. We have to work diligently and respectfully this should allow us the time to do both," he said.
Opposition Leader Bill Shorten says the country 'can do better and we have to'.
"Falling short is no reason to lower the bar," Mr Shorten said. "When it comes to closing the gap we cannot be content with aiming for anything less than proper equality."
"There is nothing to fear from recognising and owning the failures of the past," he said.
Labor MP Linda Burney pointed out very little has changed.
"We are still seeing heart disease, we are still seeing houses with up to 30 people living in them, we are still seeing Aboriginal people dying dramatically younger than non-Aboriginal Australians," she said.
In response to the Prime Minister, Mr Shorten says a future Labor Party will commit to compensation for members of the Stolen Generations.
Mr Shorten's comments were made a day before the tenth anniversary of the formal apology to the Stolen Generations delivered by Prime Minister Kevin Rudd.
Mr Shorten also committed to recognising an Indigenous voice to Parliament, recommended by the Referendum Council in the Uluru Statement from the Heart last year, after the Turnbull government rejected it.
"Who are we in this Parliament to simply reject it out of hand?" Mr Shorten asked the chamber.
"It's time for us to be better and braver than the path of least resistance."
But the government shut down the suggestions as "undesirable" and "inconsistent" with Australian democracy.
"We came to our decision after very careful consideration," Mr Turnbull told Parliament.
"The government remains absolutely committed, however, to recognising Aboriginal and Torres Strait Islander people in the Australian Constitution."
The Prime Minister flagged a new parliamentary committee to reinvigorate the process.