The US has called for an investigation into the political violence in Turkey and urged restraint on all sides following the fifth day of escalating nationwide protests against the rule of the prime minister, Recep Tayyip Erdogan.

In remarks that are likely to provoke Erdogan, the US secretary of state, John Kerry, said: "We are concerned by the reports of excessive use of force by police. We obviously hope that there will be a full investigation of those incidents and full restraint from the police force."

Earlier on Monday, Erdogan warned protesters against taking the country's political disputes on to the streets, signalling he could mobilise his mass popular support to crush the demonstrations.

Sporadic clashes between protesters and riot police continued in Izmir, Ankara and Istanbul, and two deaths were confirmed. Mehmet Ayvalitas, 20, was hit by a car on Sunday that struck protesters in the Mayis district of Istanbul, reports said.

Later Abdullah Comert, 22, died from serious gunshot wounds in the southern city of Antakya, according to the local governor's office, which said the shooter was unidentified.

Erdogan, meanwhile, lashed out at his critics, delivering an uncompromising message after a week of the worst violent turmoil of his decade in power.

Erdogan's Justice and Development party (AKP) took just over half the national vote in the last elections in 2011. Invoking that mandate, the prime minister said: "There is 50% and we can barely keep them at home. But we have called on them to calm down."

Erdogan gave similar warnings on Saturday in his first response to the turmoil that saw the centre of Istanbul turned into a battleground over the weekend as hundreds of thousands of people across the country took to the streets enraged at heavy-handing riot police handling of a demonstration last Friday.

The turmoil started a week ago as a small environmental protest in an Istanbul park against redevelopment plans, but quickly mushroomed into huge weekend clashes with riot police in the city and across half the country after police mounted tear gas and water cannon attacks on peaceful civilians. The focus of the protest shifted from the building project to the person of the prime minister.

Erdogan's dismissive attitude to the mass demonstrations contrasted increasingly with President Abdullah Gul, who sounded conciliatory and pointedly rebutted Erdogan's message. "Democracy does not mean elections alone," he said, in what appeared to be a sharp riposte to the prime minister's repeated insistence on the strength of his parliamentary mandate.

"There can be nothing more natural for the expression of various views, various situations and objections through a variety of ways, besides elections," the president said.

The prime minister has challenged the rights of the initial protesters in a central Istanbul park to launch a sit-in opposing a redevelopment scheme demolishing the green space to make way for a shopping complex, mosque, and a replica of an old military barracks.

Government data showed 1,500 arrests in Ankara, 300 in Izmir, and another 370 in the southern town of Adana. The impact of the sudden eruption of political turbulence was also felt by Turkey's business community in what has been one of the world's fastest-growing economies in recent years. Istanbul's stock exchange fell by 10.47%, the worst one-day decline in a decade. The prime minister brushed off the news. "Bourse indexes increase and decrease, they are not always stable. When we came to power, the National 100 Index was around 11,000. It might increase to 100,000 tomorrow."

An Istanbul professor and one of his students both lost eyes when police hit them in the face with tear gas canisters on Sunday night, it was confirmed.

In a further sign of spreading instability, the Turkish military said there had been an exchange of fire between its forces and Kurdish guerrillas of the Kurdistan Workers' party (PKK) in Sirnak, in the south-east. The two sides have been observing a ceasefire as the PKK stages a withdrawal from Turkey into northern Iraq as part of an incipient peace process since the start of the year. The reported clash was the first known breach of the truce. The army claim could not be corroborated.

Kerry's expression of concern about the response by the Turkish authorities was followed by a series of rebukes across Washington on Monday as the US sought to distance itself from one of its most important regional allies over the affair.

White House spokesman Jay Carney said the US had "serious concerns about the reports of excessive use of force by police and large numbers of injuries and damage to property".

The protests come at an awkward time for the US which is trying to convince the international community that governments in Syria and Iran do not respect the rights of their citizens while the Turkish-backed rebels in Syria represent a more democratic alternative.

Tensions remained high in Istanbul on Monday, with protesters still commanding the rebellion's focal point, Taksim Square. Trade unions are threatening to mobilise a general strike and several Turkish universities have postponed final examinations – a move seen as encouraging the students to take part in the wave of protest without jeopardising their academic prospects.

The prime minister blamed the sudden eruption of protest against him on unspecified foreign powers or elements. "Our intelligence work is ongoing," he said. "It is not possible to reveal their names … We shall be discussing these with them and will be following up, in fact we will also settle accounts with them."