Investigators said they have found “serious irregularities” at the federal jail in New York where Jeffrey Epstein was being held on sex-trafficking charges, deepening the mystery surrounding the disgraced financier’s death.

Mr. Epstein, 66 years old, was found dead Saturday at the Manhattan detention facility administered by the federal Bureau of Prisons. The New York City medical examiner believes Mr. Epstein’s cause of death is suicide by hanging, but is awaiting additional information from law enforcement before releasing her official findings, a city official said.

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Among the primary questions for investigators is why Mr. Epstein was taken off suicide watch and left alone with minimal supervision at the time of his death. After an earlier incident in which he was found unconscious in his cell with marks on his neck, Mr. Epstein was put in the suicide-watch unit July 23, but was removed from the watch days later at the request of his attorneys and after daily psychological evaluations.

“I was appalled and frankly angry to learn of the [jail’s] failure to adequately secure this prisoner,” said Attorney General William Barr, speaking Monday at a conference for police officers in New Orleans. “We are now learning of serious irregularities at this facility that are deeply concerning and demand a thorough investigation.”

After Mr. Epstein was removed from suicide watch, he was downgraded to “special observation status,” which mandated that guards check on him every 30 minutes and that he have a cellmate, a person familiar with his detention said.

But in the hours before Mr. Epstein’s death, his cellmate had left and was not quickly replaced as required, the person said. The identity of Mr. Epstein’s cellmate couldn’t be determined. Mr. Epstein also wasn’t receiving the mandated regular check-ins, the person said.

“We will get to the bottom of what happened, and there will be accountability,” Mr. Barr said.

The Federal Bureau of Investigation and the Justice Department’s inspector general have devoted additional manpower and resources to investigating the circumstances around Mr. Epstein’s death.

Just over a month ago, Mr. Epstein was arrested on two federal charges related to sex trafficking of minors, for which he faced a maximum sentence of 45 years in prison. He pleaded not guilty and had been awaiting trial at the detention facility after a federal judge denied his release on bail.

Prosecutors accused Mr. Epstein of orchestrating a yearslong sex-trafficking operation in which he and his associates lured dozens of girls—some as young as 14 years old—to his homes in New York and Florida to perform massages that involved sex acts.

Mr. Epstein’s death ratchets up the scrutiny of the Metropolitan Correctional Center, the facility in lower Manhattan where Mr. Epstein was housed. Inmates there have long complained about the conditions of confinement. After Mr. Epstein’s death, prison union officials said the jail suffers from staffing shortages that result in overworked guards.

“The institution is unsafe,” said Serene Gregg, the local union president that represents MCC workers. “Things are not adequately being performed.”

The Bureau of Prisons makes its own determinations about how inmates should be housed and monitored in detention facilities, according to law enforcement experts.

When Mr. Epstein was removed from suicide watch, the jail told Justice Department officials he would receive a cellmate and guards would look in on him every 30 minutes, a person familiar with the matter said.

The MCC houses nearly 800 inmates, most of whom are in pre-trial detention. Joaquín “El Chapo” Guzmán, the notorious drug-cartel leader who escaped twice from maximum-security prisons in Mexico, spent two years awaiting U.S. trial at MCC without any major incidents.

A handful of criminal prosecutions each year involves corruption among correctional officers at MCC. Last year, one officer pleaded guilty in Manhattan federal court to smuggling alcohol, cellphones and other contraband for inmates in exchange for more than $25,000 in bribes.

The investigation into Mr. Epstein’s death could take years to complete. It took more than two years, for instance, for the Justice Department’s inspector general to complete a probe into the death of Kenneth Trentadue, who was found hanging in his cell at a federal prison shortly after he was arrested on a parole violation in 1995, at the height of the Oklahoma City bombing investigation. Investigators, who began probing the death in 1997, interviewed more than 230 witnesses and reviewed thousands of pages of documents, a sign of how labor-intensive such investigations can be.

Mr. Epstein’s body has been released since the New York City medical examiner performed an autopsy Sunday.

The examiner said she needed more information before determining the cause of death. Mr. Epstein’s only known living relative is his brother, Mark Epstein.

Mr. Barr on Monday also promised the Justice Department would continue its investigation into people who allegedly helped Mr. Epstein recruit young women and girls for sex trafficking.

“Any co-conspirators should not rest easy,” Mr. Barr said. “The victims deserve justice, and they will get it.”

Federal prosecutors in Manhattan also could seek to seize Mr. Epstein’s assets through a process known as civil forfeiture. That process could allow prosecutors to sue for assets that were used to facilitate the alleged sex trafficking, including Mr. Epstein’s real estate, jets and bank accounts. If the properties are successfully forfeited, the money goes into a Justice Department fund that can be distributed to victims.

Through the civil forfeiture process, prosecutors could lay out the evidence they have gathered against Mr. Epstein that hasn’t become public, according to former federal prosecutors.

The government would be able to use investigative tools, such as search warrants and grand jury subpoenas, that wouldn’t be available to victims suing the estate, said Jaimie Nawaday, a former federal prosecutor who has handled asset forfeiture cases.

“It might be in their interest to let the government go first,” said Ms. Nawaday, now a partner at Kelley Drye & Warren LLP. “The government has a higher rate of success.”

It couldn’t be determined if Mr. Epstein had prepared a will, although estate-law experts said he likely had one given his wealth.

Write to Sadie Gurman at sadie.gurman@wsj.com and Nicole Hong at nicole.hong@wsj.com