Almost 20,000 members of the police, civil service, judiciary and army have been detained or suspended since Friday night's coup, in which more than 200 people were killed when a faction of the armed forces tried to seize power.
Inside Turkey, the main opposition Republican People's Party (CHP) also said the response must be conducted within the rule of law, adding that the plotters and those who helped them must be tried in the courts.
The spiritual leader said if he had prior knowledge of the plans to attempt overthrowing the government he would have urged plotters to reconsider.
The interior ministry said nearly 9,000 people, including nearly 8,000 police but also municipal governors and other officials, had also been dismissed in a widening purge.
Turkey purged its police on Monday after rounding up thousands of soldiers, judges and prosecutors.
Prime Minister Binali Yildirim accused the USA, which has said it will only consider an extradition request if clear evidence is provided, of double standards in its fight against terrorism.
Some were shown in photographs stripped to their underwear and handcuffed on the floors of police buses and a sports hall.
Andrew Finkel, the co-founder of P24, an initiative that supports independent Turkish media, said the pressure on journalists had "obviously gone up a notch since the [failed] coup". Akin Ozturk, alleged to be the ringleader of the July 15 uprising, and Gen. Adem Hududi, commander of Turkey's 2nd Army, which is charge of countering possible threats to Turkey from Syria, Iran and Iraq.
Speaking in Brussels, where he met European foreign ministers, including the United Kingdom foreign minister, Boris Johnson, Kerry said Turkey must provide evidence that "withstands scrutiny" when requesting the extradition of the US-based cleric Fethullah Gülen, whom Erdoğan has blamed for the attempted coup.
"The government's response to the failed coup is widening", Al Jazeera's Zeina Khodr, reporting from the capital, Ankara, said.
Ankara has demanded Washington hand Gulen over. Washington says it is prepared to extradite him, but only if Turkey provides evidence linking him to crime.
"I'm sorry but this parallel terrorist organisation will no longer be an effective pawn for any country", Mr Yildirim said.
On Sunday, Yildirim said the coup had failed and life had returned to normal, but he and other officials also urged people to take to streets at night, saying risks remained.
"Let me be very clear.no country can become an European Union state if it introduces the death penalty", she said. He was able to fly into Istanbul in the early hours of Saturday, after rebel pilots had his plane in their sights but did not shoot it down.
Addressing hundreds of supporters outside his Istanbul residence in the early hours of Tuesday, Erdogan responded to calls for the reintroduction of the death penalty with the simple statement: "You can not put aside the people's demands".
Gulen said in an interview with several media outlets including AFP at his compound in the Pennsylvania that he has "no concerns personally" about the extradition request.
However in Berlin on Monday, German Chancellor Angela Merkel's spokesman said Turkey's bid on joining the European Union would end if Ankara restored the death penalty.
Anadolu reported Tuesday that 85 high-ranking military officials have been jailed pending their trials related to the coup attempt, and dozens more are awaiting questioning. "There was a list of people suspected of planning a coup".
Turkish President Recep Tayyip Erdogan made a series of televised appearances overnight in which he disclosed dramatic details of his survival on the night of a failed coup and raised the specter of reintroducing the death penalty to punish conspirators.
Responding to the widening crackdown, the U.S. secretary of state, John Kerry, urged "the government of Turkey to uphold the highest standards of respect for the nation's democratic institutions and the rule of law".
While he recognized the need to apprehend the coup plotters, Kerry said: 'We caution against a reach that goes beyond that'.
Referring to Gulen, Kerry called on Turkey to furnish evidence "that withstands scrutiny", rather than allegations.
But EU foreign policy chief Federica Mogherini warned that such a step may end Turkey's EU membership hopes. But he said the government should "be supportive of due process and freedoms that are outlined in the Turkish constitution that include freedom of speech, freedom of press and freedom of assembly".
Inside Story - What's behind Turkey's failed coup? The remarks of Turkey's highest ranking officials, the president and prime minister, suggest the idea is being entertained. Some high-ranking military officials involved in the plot have fled overseas, he said.
Thousands of officials suspected of links to Gulen were purged from the judiciary and the Interior Ministry.