A Palestinian man tried to stab an Israeli police officer on Saturday near Jerusalem's Old City and was killed, a police spokeswoman said.
Later, the army said a Palestinian injured a soldier in the West Bank by hitting him with his car before himself being shot -- but a witness said the man had not attacked anyone before he was hit.
In the first incident, police "spotted a man who aroused their suspicions and approached him to check," the spokeswoman said. "The terrorist pulled out a knife and tried to stab the policeman and was quickly killed by police gunfire."
She said the attacker was a 26-year-old Palestinian from annexed east Jerusalem.
Meanwhile, the army said a Palestinian driver approached the Huwara checkpoint in the northern West Bank and rammed his car into security forces.
A soldier was wounded and forces responded, firing at the perpetrator, a statement said. The soldier was evacuated to hospital and the assailant was said to be "receiving medical treatment".
Palestinian sources named the driver as Maher al-Jabi, 56, who they said died of his wounds in a Nablus hospital.
A witness disputed the army claim, saying Jabi had stopped his car at the checkpoint, shown his papers and been waved through.
"But another soldier, standing on the other side of the vehicle, fired four times through the window from a very close range," the witness said.
- Clashes with security forces -
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A wave of violence since the start of October has claimed the lives of 133 people on the Palestinian side, 19 Israelis, an American and an Eritrean.
Many of the Palestinians killed have been attackers while others have been shot dead by Israeli security forces during clashes.
On Friday, Israeli border guards shot dead a Palestinian woman who tried to ram a car into them in the West Bank, shortly after a Palestinian was killed in clashes with Israeli forces near the border with the Gaza Strip.
A number of Palestinians have attempted attacks with kitchen knives in what some analysts have described as virtual suicide missions.
Palestinians have grown frustrated with Israel's occupation, the complete lack of progress in peace efforts and their own fractured leadership.
On Saturday afternoon, there were clashes with security forces after around 150 Palestinians demonstrated near the wall of the Old City in east Jerusalem to protest an Israeli policy of keeping the bodies of killed attackers rather than returning them to their families, a police statement said.
Police said they used anti-riot measures to disperse the demonstrators after some began throwing rocks.
Palestinian emergency services said some 20 people had been hurt when police fired stun grenades and rubber-coated bullets.
Pope Francis, leader of the world's 1.2 billion Roman Catholics, used his traditional Christmas Day address to urge Israelis and Palestinians to resume "direct dialogue", saying the conflict had "serious repercussions" on the Middle East.
US-backed peace talks between the Palestinians and Israel collapsed in April 2014 after nine months of fruitless meetings amid bitter recriminations and mutual blame.