One of the biggest beneficiaries of the health law’s expansion of coverage to more than 13 million people this year has been the nation’s safety-net hospitals, which treat a disproportionate share of poor and uninsured people and therefore face billions of dollars in unpaid bills.

Such facilities had expected to see a drop in uninsured patients seeking treatment, but the change has been faster and deeper than most anticipated — at least in the 25 states that expanded Medicaid in January, according to interviews with safety-net hospital officials across the country.

“This is really phenomenal,” said Ellen Kugler, executive director of the National Association of Urban Hospitals, based in Sterling, Va., which represents inner-city safety net institutions. “It shows the Affordable Care Act is clearly working in these locations.”