Donald Trump lashed out at rivals Jeb Bush and Ted Cruz overnight at the most acrimonious debate to date between Republican US presidential candidates, shouting insults and casting aside a pledge to be more measured.

The New York billionaire went into the CBS debate with a head of steam, having won New Hampshire last Tuesday and holding a big lead in polls in South Carolina a week before Republicans vote on 20 February.

Rather than play it safe, Mr Trump responded to every comment leveled his way, interrupted his opponents at will and called them liars repeatedly in an emotional outburst that could raise more questions about whether he has the temperament to serve in the White House.

He made his most blistering attacks against Mr Bush and his brother, former President George W Bush, who has many admirers in the Republican establishment.

It was hard to declare a winner in the debate amid the constant volley of insults, not all of them from Mr Trump.

Mr Cruz and fellow Senator Marco Rubio also took pointed jabs at each other over illegal immigration.

But Mr Trump dominated the debate conservation on Twitter with 40% of the mentions, according to data from the micro-blogging platform.

Mr Trump's combative style has set the tone for much of the campaign but in recent days he had pledged to pursue a more measured, positive approach.

At the debate, that strategy lasted only through his comments about Supreme Court Justice Antonin Scalia, who was found dead yesterday.

Mr Trump was quickly goaded when Mr Bush criticised his past statements that Russia has a role to play in Syria.

Russia, Mr Bush said, is not attacking Islamic State militants but instead helping Syrian President Bashir Assad, who Washington wants to leave power.

Mr Trump blasted Mr Bush's brother for launching the Iraq war in 2003 over claims, later proven false, that Iraq possessed weapons of mass destruction.

"A big, fat mistake," said Mr Trump, noting that the 9/11 attacks had also occurred on George W Bush's watch.

"George Bush made a mistake," MrTrump said. "We all make mistakes. But that one was a beauty... They lied! They said there were weapons of mass destruction. And there were none."

Many in the crowd booed Mr Trump and the Republican front-runner dismissed them as "lobbyists and special interests" supporting the former Florida governor.

Mr Bush, who has wilted in the past under assault from Mr Trump, stood firm this time. He will campaign with his brother George on Monday in North Charleston, South Carolina.

"I'm sick and tired of him going after my family," Mr Bush said. "My dad is the greatest man alive in my mind. While Donald Trump was building a reality TV show, my brother was building a security apparatus to keep us safe. And I'm proud of what he did.

"He had the gall to go after my mother," Mr Bush said, reminding the audience that Mr Trump had criticised his 90-year-old mother, Barbara Bush, wife of former President George H W Bush.

"My mother is the strongest woman I know."

"She should be running," Mr Trump responded.

Mr Bush provoked another outburst from Mr Trump by saying the Republican nominee should be someone "who doesn't brag, for example, that he has been bankrupt four times."

"That's another lie," Mr Trump said. "I never went bankrupt."