As D.C. workers and volunteers fanned out for an annual count of the city’s homeless late Wednesday, they found dozens bundled up in familiar spots around Union Station, dozens more huddled under an overpass in Foggy Bottom and others never before recorded in bushes and doorways in sight of the U.S. Capitol.

But city workers began the evening already knowing where most of the city’s homeless were: in shelters or rented motel rooms paid for by the District.

The number of people in the emergency care of the District surpassed 4,000 this week — easily on track to eclipse last year’s record-breaking number of sheltered homeless. The number has grown by more than 1,500 over the past four weeks and continues to expand daily, often with 100 families or more seeking shelter when the temperature drops below freezing, advocates say.

But most of these families are no closer to a long-term housing solution. Despite a sustained period of city revenue growth and more scrutiny than ever, the District government has failed over the past year to make serious progress on one of its most vexing and visible social challenges. Even as its new mayor, Democrat Muriel E. Bowser, promises a new course, the city is largely in a holding pattern, spending millions on emergency services and doing little that’s new to provide longer-term housing for those who need it.

Once people are given shelter — on nights when the city issues a hypothermia alert — they are almost always guaranteed a room for the remainder of winter, fueling an ever-upward climb in program numbers and cost until spring arrives.

Kristy Greenwalt, right, an expert on homeless housing policy with the city, shakes hands with Michael Jones, a man who has been homeless in the District for eight years, just after an interview and questionnaire had been completed. (Michael S. Williamson/The Washington Post)

More than 660 families are now on city shelter rolls with two full months of winter and historically on-and-off freezing weather remaining. At the current rate, the number of families in motels and shelters could top 900 by the end of March.

Bowser won office last year promising a new approach to the city’s homeless problem.

One of her most frequent applause lines on the campaign trail was to call then-Mayor Vincent C. Gray (D) “inhumane” on homelessness. She promised she would not repeat Gray’s decision to close off access to motel rooms for families at the end of January. Nor, she said, would she treat the District’s downtrodden like third-world refugees by turning, as he did, to makeshift shelters in gymnasiums with little privacy or access to showers.

When Bowser took office Jan. 2, however, the city had already reached capacity at its main shelter. To demonstrate her commitment to a new approach, she had little choice but to keep the program open to families, placing them in sought-after motel rooms until her fledgling administration could formulate a plan.

In her first month, Bowser’s staff has quietly expanded from two to four and then to six the number of motels it is using to house homeless families.

The city is now renting more than 400 rooms for the homeless each night, and at an increasing cost per room, records show.

That has raised the real possibility that the city could burn through $19 million set aside for overflow homeless services for the year and compound a budget shortfall that is already forcing cuts elsewhere. One of those cuts has been to the number of caseworkers for homeless families. Last week, Bowser froze $600,000 to hire more, which several council members questioned as short-sighted.

A hobbling homeless man makes his way on 14th street NW. Just after announcing a package of new grants to help homeless service providers across the country, top officials from HUD and other volunteers canvassed the city to count people who are in need. (Michael S. Williamson/The Washington Post)

Bowser, however, has shown little other wavering on the issue. In the clearest — and easily most ambitious — promise of her young term, Bowser on Wednesday night promised a crowd of more than 300 volunteers who turned out for the homeless count that her administration would “end chronic homelessness” in the District.

“We have a problem in our city that can be solved with the right commitment,” Bowser said. “Cities all over our nation are ending homelessness for veterans. We are going to end chronic homelessness in our city.”

Bowser stressed that her first priority would be family homelessness. But aides readily acknowledged that the obligation to provide emergency shelter to a population that keeps growing means that her team must dig out from under a crushing backlog before they can focus on her campaign promise to close the city’s troubled shelter on the old D.C. General Hospital campus.

The backlog also could delay wholesale changes to what most agree is a broken system predominantly serving a population of young, poor, single mothers living on the margins.

Because the city offers emergency shelter to families only on freezing nights, it faces a mountain of pent-up demand for social services at the start of each winter. Among those flooding the system are women and children seeking to escape abusive partners or parents as well as those who have worn out their welcome with friends or family members — including some hosts who put their own leases or public-housing subsidies at risk by letting families double up.

Gray’s administration estimated last year that at any time, about 5,000 families in the District are on the brink of homelessness, sharing apartments with others or floating among multiple addresses as parents lose income or housing of their own.

During Wednesday’s count, Bowser said the city’s spike in homelessness and lack of affordable housing are connected — and impossible to solve independently.

“None of us can be proud of how our city is growing and progressing,” she said, “when so many people are left out of that progress.”

Bowser did not walk the streets with reporters, as Gray had done in recent years, but members of her new team — each highly respected among homeless advocates — did.

Part of one group scouring Pennsylvania Avenue SE was Laura Zeilinger, former executive director of President Obama’s U.S. Interagency Council on Homelessness and the new leader of the embattled Department of Human Services; Polly Donaldson, Bowser’s new lead on housing and community development, who has been on the front line of the issue in the District for years; and Kristy Greenwalt, another national expert on homeless housing policy who joined the city last year.

Mixed in was Shaun Donovan, head of the White House Office of Management and Budget, and Ann Marie Oliva, a director with the Department of Housing and Urban Development — both on their sixth homeless count since Obama took office.

After 10 p.m., Zeilinger and Donovan got a firsthand look at another side of the problem: homeless single adults, many of whom struggle with mental illness and refuse help.

In the entranceway of an Edible Arrangements storefront on Capitol Hill, Zeilinger squatted to greet a man wrapped in blankets and surrounded by a wall of trash bags full of coats and clothes he’d hoarded.

“Hi, I’m Laura,” she said. “How long have you been out here?” The man looked up and asked: “Laura who? Laura Bush?”

“Just call me Laura Z.,” she said.

A couple blocks later, it was Donovan’s turn. He asked a man wearing a Detroit Tigers hat and shifting his weight in the entrance to a darkened nail salon if he could help.

“You look like R2-D2,” the man said.

Eventually, both offered their names but declined rides to a shelter, where so many already were.

Greenwalt and Oliva were among the last to make a final round near 1 a.m., and by then, the four people camped in front of Edible Arrangements were bundling up for sleep. A mile to the north, another team persuaded a man and a woman to accept a ride to shelter.

Zeilinger, who is scheduled to testify for the first time Friday before the D.C. Council, cautioned that there is no quick fix for the challenge the city faces on homelessness. The city can only begin to stabilize the population when the District’s stock of affordable housing rises, she said.

“This is not a problem that just happened,” Zeilinger said. “We have a homeless system that tends to expand, expand, but we don’t have a supply of affordable housing that has kept pace with the need.”

Zeilinger, who began her new job Monday, said she wished there was another path for the city besides motels this year, but the administration was not willing to cut off families being sent there. Instead, officials will work with families as winter ends to try to ease them into more permanent housing.

“We certainly don’t see motels as the long-term strategy on homelessness,” she said. “They are not a good use of taxpayer dollars, and every dollar we spending on motels are dollars we are not spending on housing. That said, we are not going to be able to make that transition overnight. We are not going to pull the rug out from under anybody.”

Juliet Eilperin contributed to this report.