A second person has died from injuries sustained during overnight clashes between demonstrators and the police in the Istanbul neighbourhood of Okmeydani, prompting fears of further violence before the anniversary of anti-government protests a year ago.

Tensions were further stoked on Friday when the prime minister, Recep Tayyip Erdoğan, gave a characteristically pugnacious speech, in which he defended police who shot and killed a bystander during protests on Thursday.

The Istanbul governor, Huseyin Avni Mutlu, said Friday's victim was one of two people wounded by shrapnel from a hand grenade, and called for calm "for the security of the nation".

"Everyone must help bring the situation back to normal. We need calm, we need to act in a calm manner for the security of Istanbul and of the nation," Mutlu told reporters, warning residents to be on their guard against "provocations".

The Turkish daily Hürriyet reported that the explosive device had detonated during street battles between protesters and police on Thursday night, injuring one civilian and at least seven police officers. None of them are said to be in critical condition.

Friday's death follows the fatal shooting a day earlier of a 34-year-old man who was hit by a stray bullet while attending a funeral in the nearby Okmeydani Cemevi, an Alevi place of worship.

Ugur Kurt, a cleaner, was killed when the funeral procession passed near a group of demonstrators who had gathered to commemorate the death of Berkin Elvan, a 15-year-old who was killed by a teargas cartridge during last summer's anti-Erdoğan protests.

Witnesses said that the bullet was fired by a policeman after protesters threw Molotov cocktails at an armoured police vehicle.

Speaking at a meeting of his Justice and Development party (AKP) on Friday morning, Erdoğan said: "We were silent while mourning the 301 dead [from the Soma mine disaster]. But excuse me – we will not be silent any longer. We will not be silent in the face of those who use the excuse of Soma to vandalise. So allegedly they staged a commemoration for Berkin Elvan. Should we stage a commemoration for everyone who dies? He is dead and gone."

Erdoğan also defended the police. "A Molotov cocktail fell into a police vehicle. [The officers] suffered burns. They are being treated in a hospital. These terrorists try to break the windows of all of these vehicles. Should the police just stand by and do nothing? I don't understand [the officers'] patience. The media never write about these wounded police officers."

The firearms of 20 police officers who were on duty at the time of Kurt's shooting were sent for a ballistic examination as part of the investigation into the death, authorities said on Friday. But it took more than 24 hours for a state prosecutor to attend the scene of Kurt's death.

Efkan Bolaç, a lawyer present at the Cemevi, told Turkish media: "There was a shooting incident. Evidence needs to be collected. Even if the prosecutor doesn't come, the police should have been dispatched to collect evidence."

Okmeydani is home to a community of Alevis, a religious minority in mainly Sunni Muslim Turkey who espouse a liberal version of Islam and have often been at odds with Erdoğan's socially conservative, Islamist-rooted government.

It has seen frequent leftwing protests in the past.