The Israeli parliament called a special session for Monday to discuss the recent wave of attacks while the country’s army said it was deploying additional combat troops in the occupied West Bank.
The Palestinian presidency meanwhile blamed the “environment [of] tension” on a “dangerous” escalation by Israel as well as Mr Trump’s deal.
Israeli Prime Minister Benjamin Netanyahu vowed to find those responsible for the violence, adding “terrorism will not defeat us, we will win.”
Last Tuesday, flanked by Mr Netanyahu, Mr Trump unveiled his “deal of the century” that analysts have called the most pro-Israel solution to the decades-old conflict to come out of the White House.
The 181-page proposal allows Israel to annex swathes of territory in the occupied West Bank as well as Jerusalem, land which the Palestinian had hoped would make their future state.
It was roundly rejected by the Palestinian leadership and has triggered tensions across the region.
On Thursday Israeli police said they shot dead an Arab citizen of Israel who opened fire at forces near Lion’s Gate Jerusalem's Old City, lightly wounding an officer.
Earlier in the day also in Jerusalem, a Palestinian motorist slammed his car into a group of Israeli soldiers, wounding 12 including one severely, before fleeing the scene, according to the Israeli military.
Meanwhile the occupied West Bank, an assailant opened fire on an Israeli soldier lightly injuring him near Qalqilya.
In Jenin, a Palestinian police officer Tarek Badwan, 25, died after being critically injured by Israeli army gunfire, the Palestinian ministry of health reported.
Also on Thursday, Israeli soldiers shot dead a 19-year-old during a protest against the Israeli demolition of the home of a Palestinian who was involved in the 2018 killing of a Jewish settler.
An Israeli military spokesman said the young man was throwing rocks.
Just the day before, Israeli troops killed a 17-year-old Palestinian also in the West Bank, saying he had thrown a fire-bomb at them during a separate violent rally against Trump's plan. He was the first fatality since the plan was announced.
Nabil Abu Rudeineh, a spokesman for Palestinian president Mahmoud Abbas blamed the Israelis and the deal for the troubles saying they were both “trying to impose fake facts on the ground."
Mr Netanyahu vowed to punish any perpetrators behind the attacks, adding it was " just a matter of time" until they find the person responsible for the car-ramming incident.
Lt. Col. Jonathan Conricus, the military spokesman, said Israeli forces are "not trying to escalate the situation while understanding the complexity and sensitivity of the situation."
He stopped short of linking the violence to Trump's plan.
The long-awaited proposal, largely authored by Mr Trump's son-in-law Jared Kushner, has been met with mounting criticism with many expressing fears it gives Israel a green-light to begin annexing Palestinian land even before it is signed.
Israel has backed the plan, alongside some European states such as Hungary and the UK.
However, the European Union has rejected parts of it alongside the Arab League and the 57-member Organisation of Islamic Cooperation, including Saudi Arabia.
Tensions have soared after Mr Netanyahu promised to speed ahead with annexing the strategic Jordan Valley. However, facing some pressure from the US administration, the Israeli premier appears to be holding off until after March elections.
It has also triggered a flare-up of hostilities between Israel and the blockaded Gaza enclave.
Over the last few days, Palestinian militants have launched mortar fire, rockets and balloon-borne explosives into Israel, causing panic but no serious casualties. Israel has carried out nightly strikes against sites belonging to Gaza's ruling Hamas Islamists.
On Thursday Hamas welcomed the violence in the West Bank.
"The spreading resistance and clashes by our people in the West Bank and their resistance in the heart of occupied Jerusalem is an active response against the destructive Trump deal," Hamas spokesman Hazem Qassem said.