More than 60% of Palestinians in the West Bank and Gaza want Mr. Abbas to resign, according to a public-opinion poll released Wednesday by the Palestinian Center for Policy and Survey Research.

Still, 64% of Palestinians say they worry that if Mr. ​Abbas leaves politics, it could spark turmoil and insecurity because he has no clear successor.

The president is now in the 13th year of what initially was meant to be a four-year term and has been in and out of the hospital in recent months. He shows no signs of relinquishing power.

A spokesman for Mr. Abbas didn’t respond to questions about criticism of his leadership. Aides defended Mr. Abbas as an important figure who has helped shape Palestinian institutions.

But some acknowledged that progress toward a deal with Israel and improvements in daily life haven’t come fast enough.

“We have never reached this level of weakness in front of our people,” said Saeb Erekat, a 63-year-old top aide to Mr. Abbas.

“At the end of the day, governing is not about how decent you are and how honest you are,” Mr. Erekat said. “You promised your people you do something. You didn’t.”

Crumbling public services have generated frustration. Mounds of garbage piled up on the streets of Ramallah last month during a strike by sanitation workers. At the Palestine Medical Complex in the city, patients complain about poorly trained medical personnel and inadequate equipment.

“You can’t give security all the budget and leave the patient with nothing,” said Atef​Abu Qubita, 47, who was getting treatment for diabetes.

Mr. Abbas’s fragile position is adding to skepticism about the success of any peace plan advanced by the Trump administration.

Washington has described Mr. Abbas as a potential spoiler. Palestinian officials have boycotted the Trump administration since its December announcement it was moving the U.S. embassy to the contested city of Jerusalem.

In a recent interview with a Palestinian newspaper, Jared Kushner, President Donald Trump’s son-in-law and senior adviser, criticized Mr. Abbas, saying: “There are a lot of sharp statements and condemnations, but no ideas or efforts with prospects of success.”

In response, Abbas spokesperson Nabil Abu Rudeineh said the Palestinian leader “cannot be bypassed, neither regionally nor internationally.”

The territories of the West Bank, governed by the Palestinian Authority, and the Gaza Strip, run by Hamas, are divided. Ordinary Palestinians are frustrated with the failure of the government to deliver basic services amid high poverty and unemployment.

“Unfortunately, our experience with democracy was a big failure,” said Qassam Barghouti, who works in public relations. Palestinian leaders have failed to make “space for young people and for women to be well-represented in the decision-making circles in the country.”

Mr. Barghouti is the son of Marwan Barghouti, a politician viewed as a possible successor to Mr. Abbas who is now in prison in Israel after being convicted on murder charges in connection with a Palestinian uprising known as the Second Intifada.

Officials say time isn’t on their side and that they fear more radical leaders, backed by restless young people, could end up in charge.

About 70% of Palestinians in the West Bank and Gaza are 29 and younger, according to the Palestinian Bureau of Statistics. Those who run the Palestinian Authority are in their 60s, 70s and 80s.

Under the 1993 Oslo Accords that set the terms for relations between Israel and the Palestinians, the Palestinian Authority was meant to be an interim government until a sovereign state could be established. Palestinian officials say they have been unable to hold elections while the West Bank and the Gaza Strip are divided.

A recent reconciliation effort has stalled, as Hamas, which controls Gaza, has refused to dissolve its armed wing. There are also financial disagreements.

Palestinians have directed fresh ire at Mr. Abbas because he has slashed government salaries in the Gaza Strip, a move also deeply unpopular in the West Bank.

The Palestinian Authority controlled the Gaza Strip until 2007, and had continued to pay its employees there even after Hamas took control. Mr. Abbas hoped the cuts would pressure Hamas in a stalled reconciliation process, but the hardship has deepened.

Last month, security forces broke up a demonstration in Ramallah over the cuts, and a protest several days earlier drew 2,000 people.

Fadi Quran, 30, one of the organizers, said activists limited the focus to Mr. Abbas’s Gaza policy rather than calling for his resignation, since many aren’t ready to cross that line.

“People don’t have a vision of what happens next,” he said.

Write to Felicia Schwartz at Felicia.Schwartz@wsj.com