In a late-night spray on social media platform Truth Social, Mr Trump slammed Mr Turnbull on Monday afternoon (AEDT).
"Malcolm Turnbull, the former Prime Minister of Australia who was always leading that wonderful country from "behind," never understood what was going on in China, nor did he have the capacity to do so," the president wrote.
"I always thought he was a weak and ineffective leader and, obviously, Australian's agreed with me!!!"
Mr Turnbull lost the prime ministership in a party leadership spill and was not voted out by the public.
Mr Turnbull hit back at the president on Monday night, telling ABC TV that Australia had to be realistic about the impact Mr Trump was having "on the world, on the Western alliance, on markets, on our economies".
The former prime minster warned against Australia joining a "conga line of sycophants creeping through the White House, paying homage to this guy, telling him he's a genius".
"The reality is, if you suck up to bullies, whether it's the global affairs, or in the playground, you just get more bullying," Mr Turnbull said on the 7.30 program.
"And unfortunately, we are now seeing somebody that is utterly unconstrained, and if the advice is to go and suck up to him, where does that get you?"
He said China would take advantage of Mr Trump's "erratic" behaviour.
"What China will do is they will say they'll be the opposite of Trump: where Trump is erratic, they'll be consistent; where Trump is rude, they'll be respectful.
"They'll play a very different game to what they did in the first Trump administration."
Mr Turnbull had previously said the US president was unpredictable and could not be trusted to take military action if Australia were attacked.
He said on Monday night he also did not believe the president would go to war with China over Taiwan.
Mr Trump has not said whether he will exempt Australia from the tariffs, with a decision expected within days.
Prime Minister Anthony Albanese said his government would continue to engage "constructively" with the Trump administration.
"We are partners with the United States through our free trade agreement, and we'll continue to advocate for Australia's national interest," he told reporters in Lismore before the president's social media post.
Australia is arguing it should be handed an exemption because it imports more from the US than what it exports, the same successful argument Mr Turnbull used when he was prime minister during Mr Trump's first term.
The US president's top economic advisor Kevin Hassett has said he doubted any exceptions would be made.
Mr Turnbull told 7.30 the government was "rinsing and repeating" the arguments of 2018, but the situation had changed because he believed Mr Trump had regretted the exemption because he then had to grant exemptions to other countries.
