President Erdogan has been warned not to use the failed coup in Turkey as a “blank cheque” to bypass democratic principles.
Six thousand people have been arrested over the uprising amid growing fears the government will use the attempted putsch as a cover for cracking down on legitimate dissent.
High-ranking soldiers and 2,700 judges are among those to have been detained following clashes in which 265 people were killed.
President Recep Tayyip Erdogan has said that parliament will consider introducing the death penalty to deal with those believed to have committed traitorous acts against the state.
Mr Erdogan spoke to his supporters in front of his Istanbul residence on Sunday evening.
His speech was punctuated by frequent calls of "we want the death penalty" from the large crowd, to which Mr Erdogan responded: "We hear your request. In a democracy, whatever the people want they will get."
Adding that they will be in contact with Turkey's opposition parties to reach a position of capital punishment, he said: "We will not delay this decision for long. Because those who attempt a coup in this country must pay."
Turkey has not executed anyone since 1984 and capital punishment was legally abolished in 2004 as part of its bid to join the European Union.
And Turkey's prime minister, Binali Yildirim, said that life was returning to normal and urged supporters to remain on the streets to show their support for the government.
He also issued a stark warning to opponents: “Another calamity has been thwarted. However, our duty is not over. We shall rapidly conduct the cleansing operation so that they cannot again show the audacity of coming against the will of the people.”
Addressing tens of thousands on the sreets who were chanting “We want the death penalty! We want the death penalty!”, Mr Yildrim told protestors: “We got your message. The necessary will be done.”
Speaking to television channel France 3, France’s foreign minister Jean-Marc Ayrault urged Turkish leaders to rule within the boundaries of the law.: “[The coup] is not a blank cheque for Mr Erdogan. There cannot be purges, the rule of law must work.”
He added European ministers meeting on Monday in Brussels would reiterate that Turkey had to conform to Europe’s democratic principles.
The coup, which was launched on Friday night by members of the military, has also increased tensions with the US, with Turkey's leader demanding the extradition of a US-based cleric accused of orchestrating the violence.
On Sunday, President Erdogan vowed to “clean all state institutions of the virus” of Fethullah Gulen supporters.
He said Turkey, through the justice ministry and foreign ministry, would request the extradition of the cleric, who is based in the United States, and his backers.
Earlier, Mr Yildirim warned Barack Obama that “any country that stands behind him is no friend of Turkey, is engaged in a serious war with Turkey”.
“Today, after this coup attempt, I’m once again calling on you, I’m saying: extradite this man in Pennsylvania to Turkey now,” he added.
Suleyman Soylu, Turkey's labour minister, went further than Erdogan, suggesting the US was behind the coup. “The US is behind the coup attempt. A few journals that are published there [in the US] have been conducting activities for several months. For many months we have sent requests to the US concerning Fethullah Gulen. The US must extradite him,” he said.
At a news conference on Saturday in Saylorsburg, Pennsylvania, Mr Gulen strongly denied any role in or knowledge of the coup.
“Government should be won through a process of free and fair elections, not force,” he said.
“As someone who suffered under multiple military coups during the past five decades, it is especially insulting to be accused of having any link to such an attempt. I categorically deny such accusations.”
US Secretary of State John Kerry said the United States would entertain an extradition request for Mr Gulen, but Turkey would have to present legitimate evidence that withstands scrutiny.
The country’s Justice Minister, Bekir Bozdag, has also warned that the number of those likely to be arrested will continue to rise. “The clean-up operations are continuing,” he said.
On Sunday, the commander of the Incirlik Air Base in southern Turkey that is used by the US and other coalition partners was detained for complicity in the attempted coup, a government official said.
Incirlik is used by the United States and other coalition partners in the fight against Isis militants in Syria.
Those air operations were blocked, at least temporarily, by the Turkish government following the coup, but the US secretary of state John Kerry said operations had not been disrupted.
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