Flabbergasted by recent reports of San Francisco General Hospital sending huge bills to some privately insured patients, a Board of Supervisors committee held a hearing Thursday to determine how the city can treat insured patients more fairly.

The Department of Public Health is working on ways it can add more transparency to its billing practices and presented some of those findings to the Government Audit and Oversight Committee. The hearing came one week after Mayor London Breed and the department halted the practice of balance billing — sending patients bills if their private insurance companies don’t pay the charges — for 90 days.

At issue is San Francisco General’s policy of not entering into contracts with private insurance companies. Instead, the hospital charges the maximum rates approved each year by the mayor and Board of Supervisors. The patient’s insurance company decides how much it wants to pay, with the patient billed for the rest. That has left patients with exorbitant bills — some that reach into the tens of thousands of dollars — for common ailments like broken bones or appendicitis.

“As we’ve had more attention on this issue, it is clear that this practice does not reflect our values and the way we want our patients to experience the hospital,” said Greg Wagner, chief financial officer for the Department of Public Health, which oversees the hospital.

Wagner said DPH is considering a cap on out-of-pocket expenses that a lower-income, insured patient would have to pay. Other proposals include new administrative procedures on how to assess patient eligibility for financial support before they are billed and increasing the hospital’s financial assistance programs.

Supervisor Aaron Peskin called for the hearing after numerous reports on Vox and in The Chronicle revealed details of the hospital’s billing practices.

“Balance billing is part of a larger issue of what is, quite frankly, a screwed-up patchwork of national health care coverage,” Peskin said. “We have to take the patients out of the middle.”

Since the issue has come to light, Peskin said his email inbox has been flooded with reports of people receiving bills for tens of thousands of dollars, despite the fact that they have insurance.

One of them was Nicki Pogue, a 50-year-old who was hit with a $13,700 bill after going to the hospital for a viral syndrome. Her insurance only covered about $3,600, leading her to believe that she would have been better off if she hadn’t been insured.

“During my intake, I was never informed that I would have received a bill like this,” she said. “I received excellent care there, but its billing practices are an embarrassment for our city.”

San Francisco General has defended its billing practices by saying many of its patients are on Medicare or Medi-Cal, or have no insurance at all. A hospital spokesman told Vox, “We feel like we have to recoup what we’re able to from people who are insured because we’re supporting people who don’t have insurance.”

Wagner pushed against the comment in Vox on Thursday. While he admitted the hospital needs to re-evaluate how it deals with insured patients, he said the department does not “charge patients rates to subsidize anyone else.”

The department will present the board with more specific solutions in the next few months.

In the meantime, people like 32-year-old Michal Kaczmarski are stuck fighting bills that they worry will throw them into debt.

In August, Kaczmarski went to the hospital in the middle of the night with a debilitating stomachache. After a number of blood tests, he was told to rest and take some Tylenol. Then he received a $9,000 bill — only $2,500 of which his insurance would cover.

“Had I known that I was going to a place that charges a crazy amount, I wouldn’t have gone,” he said.

Trisha Thadani is a San Francisco Chronicle staff writer. Email: tthadani@sfchronicle.com Twitter: @TrishaThadani