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The family of a Turkish-American woman shot dead while demonstrating against Israeli settlements in the occupied West Bank demanded an independent investigation into her death on Saturday, accusing the Israeli military of killing her "violently".

Aysenur Ezgi Eygi, 26, was "shot in the head" while participating in a demonstration in Beita in the West Bank on Friday, the United Nations rights office said.

"Her presence in our lives was taken needlessly, unlawfully, and violently by the Israeli military," Eygi's family said in a statement.

"A US citizen, Aysenur was peacefully standing for justice when she was killed by a bullet that video shows came from an Israeli military shooter.

"We call on President (Joe) Biden, Vice President (Kamala) Harris, and Secretary of State (Antony) Blinken to order an independent investigation into the unlawful killing of a US citizen and to ensure full accountability for the guilty parties."

The Israeli military said its forces "responded with fire toward a main instigator of violent activity who hurled rocks at the forces and posed a threat to them" during the protest.

Eygi was a member of the International Solidarity Movement (ISM), a pro-Palestinian organisation, and was in Beita on Friday for a weekly demonstration against Israeli settlements, according to ISM.

The group on Saturday dismissed claims that ISM activists threw rocks at Israeli forces as "false" and said the demonstration was peaceful.

"Aysenur was more than 200 metres away from where the Israeli soldiers were, and there were no confrontations there at all in the minutes before she was shot," ISM said in a statement.

'Tragic' death

In recent years, pro-Palestinian demonstrators have frequently held weekly protests against the Eviatar settlement outpost overlooking Beita, which is backed by far-right Israeli ministers.

During Friday's protest, Eygi was shot in the head, according to the UN rights office and Rafidia hospital where she was pronounced dead.

Turkey said she was killed by "Israeli occupation soldiers", with President Recep Tayyip Erdogan condemning the Israeli action as "barbaric".

Washington called it a "tragic" event and has pressed its close ally Israel to investigate.

But her family has demanded an independent probe.

"Given the circumstances of Aysenur's killing, an Israeli investigation is not adequate," her family said.

On Saturday, footage provided by news agency AFP showed Eygi's body, wrapped in a blue cloth, kept in a morgue next to the body of a young girl who was killed the previous day in a separate incident in the West Bank.

The Palestinian health ministry said the 12-year-old Palestinian girl was shot and killed by "occupation (Israel) bullets" in Qaryut, near Beita.

On Saturday, Nablus governor Ghassan Daghlas accused Israeli forces of killing the two.

"Both were killed by the same bullets.... The same bullets," he said, referring to Israeli forces.

"We call out the international community to stop the insane war on Palestine. Bullets do not differentiate between activists and a Palestinian child," he said.

Eygi's family said she always advocated "an end to the violence against the people of Palestine".

Israeli settlements in the West Bank -- where about 490,000 people live -- are illegal under international law.

Since October 7, Israeli troops or settlers have killed at least 662 Palestinians in the West Bank, according to the Palestinian health ministry.

At least 23 Israelis, including security forces, have been killed in Palestinian attacks during the same period, according to Israeli officials.