One thousand extra police officers have been sent to the city, and dozens of Palestinians have been arrested.
Israeli police say three soldiers have been wounded, one of them seriously, after a Palestinian drove his car into them on a road near Hebron in the West Bank.
Police spokeswoman Luba Samri said the driver of the “large commercial vehicle” bearing a Palestinian licence plate fled the scene.
The incident came hours after one Israeli was killed and 14 others were injured in a similar attack in Jerusalem.
“After the car came to a halt, the driver, who was wounded, got out of the vehicle and started to hit people with an iron bar,” the police spokeswoman said about the first attack. He was shot by police who were in the area at the time, she said.
Police described the Jerusalem incident as a “hit-and-run terror attack” and said it took place in the same area as a similar attack two weeks ago, in which a Palestinian man rammed a car into a crowd killing a woman and a baby.
Wednesday’s incidents came amid continued tensions over right-wing Jewish demands to be able to pray inside Jerusalem’s holiest compound and the expansion of Israeli settlement building in East Jerusalem.
The site, in which Al-Aqsa mosque is located, is the holiest in Judaism, and is called Temple Mount by Jews because of the Jewish temples that stood there in biblical times. Muslims call it Haram al-Sharif, or Noble Sanctuary, and it is their third-holiest site after Mecca and Medina in Saudi Arabia.
Series of confrontations
In Jerusalem, Israeli security officers clashed with Palestinian protesters after far-right Israelis tried to storm the Al-Aqsa mosque in the first of a series of confrontations.
“There is still a lot of tension and a huge police presence here,” Al Jazeera’s Imtiaz Tyab reported from Jerusalem.
Israeli security forces briefly entered the Al-Aqsa compound as clashes broke out between Palestinian protesters and far-right Israelis trying to enter the compound.
“In a rare move, we understand that Israeli security forces entered the mosque at Al-Aqsa,” our correspondent said.
Israeli police told Al Jazeera that they had only entered briefly to close the door, but our correspondent said the move was likely to heighten tensions in the Israeli-occupied East Jerusalem.
“These provocations, as it’s seen by Palestinians at the Al-Aqsa mosque, will certainly not do anything to calm the situation.”
Israeli police said protesters threw stones and firecrackers at police officers minutes before the mosque compound opened.
“Police entered the area, pushed the masked rioters back, and they fled back into Al-Aqsa. Police closed the front gate of the mosque but did not enter,” Micky Rosenfeld, Israeli police spokesman, said.
Daily clashes between Israeli security forces and Palestinians in the streets of East Jerusalem and the Old City compound have stoked fears of a new Palestinian Intifada, or uprising.
The Al-Aqsa mosque is managed by Jordan as part of a 20-year-old peace deal.
On Tuesday, Jordan recalled its ambassador in protest at what it calls “unprecedented Israeli escalation” in Jerusalem, Al Jazeera’s Nisreen El-Shamayleh, who is in Amman, reported.
She also said that Jordan was filing a formal complaint against Israel before the UN Security Council on the Al-Aqsa incident.
Al Jazeera’s Tyab said the Israeli action, including the use of stun grenades to disperse the crowd, amounted to “a crackdown in occupied East Jerusalem”.
Omar Alkeswani, a Palestinian manager of Al-Aqsa, said police entered the compound and that 20 people were wounded in the clashes.