The prime minister was speaking in London at the Policy Exchange think tank when he declared the Liberal Party is not conservative and must work to remain the "sensible centre", noting it was Abbott who first uttered those words.
Liberal senator Eric Abetz, one of the party's most prominent conservatives, defended Mr Turnbull's speech and said "hysterical media have chose to dishonestly spin the speech in such a way to inflame tensions".
"The three of us did our best not to burst into tears", he said.
Mr Turnbull, who received an intelligence and security briefing in London on Monday, told reporters at a joint media conference with British leader Theresa May later he was "always interested in learning about the British experience".
Malcolm Turnbull's cabinet has rallied behind his definition of the Liberal Party. "It's why the Australian public have trusted us more often than not with the government of the country over the socialism of the Labor Party".
"I'm at a loss to explain why Malcolm would give that speech, unless he is just trying to further alienate the so-called conservatives from the party", he said.
Mrs May said she and Mr Turnbull had discussed cooperation in the fight against terrorism and in denying the use of the internet as a safe space for extremist groups.
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Mr Turnbull said Sir Robert "went to great pains" not to call his party conservative, adding that the founder had been determined to create a progressive party.
"What it's sent is a very strong message to conservatives in Australia [that] you're no longer welcome within the party of Menzies", Senator Bernardi told 7.30 on Tuesday night.
"There are no two nations in the world that trust each other more than Australia and the United Kingdon", Turnbull said.
He said he agreed with Mr Turnbull that "the sensible centre is the place to be".
Meanwhile, former Liberal Premier of New South Wales, Nick Greiner compounded the criticism of former PM Abbott in saying that Abbott's "five point plan" to win the next federal election, which included plans to reform the Senate to make it easier for the governing party to pass legislation, was "never going to happen".
Opposition Leader Bill Shorten said Mr Turnbull was engaging in navel-gazing. But what is beyond doubt is that he is not a leader.
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