Israeli police have arrested six Jewish suspects in connection with the abduction and murder of a Palestinian teenager in East Jerusalem last week.

The prime minister, Binyamin Netanyahu, swiftly issued a statement denouncing Jewish extremism as "terrorism" after the arrests in the early hours. "We do not distinguish terror from terror," he said.

The suspects, including some minors, are being held by the Shin Bet security agency. They are being represented by lawyers hired by a rightwing legal action group.

The news of the arrests broke after Israeli media reported that police had made a breakthrough in the investigation and had concluded that the killing was "nationalist" in motivation rather than criminal.

"Apparently the people arrested in relation to the case belong to an extremist Jewish group," an official said, speaking on condition of anonymity.

Palestinians have insisted for days that Mohamed Abu Khdeir was the victim of far-right Jews avenging the killing of three Israeli teenagers who disappeared while hitchhiking in the occupied West Bank on 12 June and whose bodies were found on Monday. Israel blames Hamas for the deaths of Naftali Frankel and Gil-ad Sha'er, both 16, and Eyal Yifrach, 19.

The killings have prompted days of riots and raised concerns of a new Palestinian uprising, or intifada.

Speaking after visiting the parents of the Israeli teenagers, Netanyahu said: "I would like to send my condolences to the Abu Khdeir family. I pledge that the perpetrators of this horrific crime will face the full weight of the law. I know that in our society, the society of Israel, there is no place for such murderers. And that's the difference between us and our neighbours. They consider murderers to be heroes. They name public squares after them. We don't. We condemn them and we put them on trial and we'll put them in prison."

Israel said its aircraft attacked 10 sites in Gaza overnight on Saturday in response to multiple rocket strikes on southern Israel.

Hamas's armed wing said seven of its fighters had been killed in a separate Israeli airstrike.

Following news of the arrests, the Palestinian president, Mahmoud Abbas, sent a letter to the UN secretary general, Ban Ki-moon, asking the UN to investigate "terror crimes conducted against our people, including the burning alive of Mohamed Abu Khdeir".

Few concrete details of the arrests were available, with Israeli media reporting on the case still covered by a gag order.

Sources said some of those arrested had been involved in a violent demonstration on the evening prior to the killing organised by La Familia, a racist group of supporters of Beitar Jerusalem football club. Participants in the protest chanted "death to Arabs" and attacked Palestinian passersby. Some of those arrested are believed to be known to police and to have criminal records.

The father of the murdered boy, Hussein Abu Khdeir, complained that he had been told nothing by police.

According to a report on Israel's Channel 2, the suspects are being investigated for alleged links to two cars, one believed to have been used in the Abu Khdeir killing and in the attempted kidnapping of another boy, Musa Zalloum, two days before.

According to the broadcaster, days before Abu Khdeir's murder, some of the suspects allegedly scouted the area in East Jerusalem where the teenager lived, attempting unsuccessfully to kidnap Zalloum.

Footage has emerged showing the faces of possible suspects in the abduction. The footage, acquired by the website Electronic Intifada, which champions Palestinian issues, appears to replicate video shot from another angle that was acquired by the Guardian on Friday and which Abu Khdeir's family says depicts the moment of his abduction.

The new footage, from a CCTV camera apparently covering a spot close to where Abu Khdeir was sitting at 3.45am on Wednesday waiting to pray, shows a much smaller part of the action. The same actions are visible at the same time as in the other video, including one man gesticulating sharply with his arm.

In the first footage it appeared as though the man was asking for directions; in the latest footage he appears to be jabbing his arm angrily at the person he is speaking to out of shot. The men in the video appear well-dressed and in their late teens to early 20s, with short dark hair.

A senior Palestinian official claimed Abu Khdeir was burned alive after suffering a head injury. The allegation is said to be based on initial postmortem findings that discovered soot deposits in his lungs, suggesting he was breathing when he was set on fire. "The direct cause of death was burns as a result of fire and its complications," the Palestinian attorney general, Mohammed al-A'wewy, said on Friday. Israeli officials have yet to release their postmortem findings.

Saber al-Aloul, director of the Palestinian forensic institute, attended the postmortem, which was carried out by Israeli doctors in Tel Aviv. A'wewy said Aloul reported that fire-dust material had been found in Abu Khdeir's respiratory canal, which meant "the boy had inhaled this material while he was burned alive". Burns covered 90% of the victim's body.

On Saturday, protesters in northern Arab towns threw stones at passing cars, burned tyres and hurled fire bombs at police, who responded with teargas and stun grenades. More than 20 people were arrested.

In a separate incident, it was claimed by relatives of Abu Khdeir that his 15-year-old cousin Tariq, a US citizen who goes to school in Florida, was beaten by police during clashes on Thursday, a day before Abu Khdeir's funeral. Tariq's parents, Suha and Salah, said he was detained and treated at an Israeli hospital. The US state department said it was "profoundly troubled" by the reports and demanded an urgent investigation.

An Israeli police spokeswoman said Tariq had attacked police and resisted arrest. He was detained, with a slingshot in his possession, along with six other protesters, including some armed with knives, she said. Tariq's father said he witnessed his son's arrest and insisted the boy was not involved in the violence, adding that several officers were hurt in the protest, one of many that day.