Senate Republicans and Democrats linked arms and voted to push the first overwhelmingly bipartisan border bill in years through committee Wednesday, giving President Trump more money than he asked for to deal with the migrant surge — but changing how he wanted it spent.

The $4.6 billion measure pumps cash into the Department of Health and Human Services so it can continue to care for children who arrive at the border without parents and who have come in such large numbers that they have drained the government dry.

The bill gives Mr. Trump more than he asked for Customs and Border Protection, plumping up funds to process immigrants at the border and to relieve overcrowding in facilities designed to hold a few dozen people but now regularly hold hundreds.

But under Democrats’ prodding, Republican lawmakers deleted money Mr. Trump had requested to increase detention capacity in the interior of the country, leaving the Department of Homeland Security with a tricky decision later this year about letting illegal immigrants be released into the community.

The goal, all sides said, was to get a bill passed quickly, and that meant compromising.

“The situation is past the breaking point. We must act,” said Sen. Richard C. Shelby, Alabama Republican and chairman of the Senate Appropriations Committee.

The measure cleared his panel on a 30-1 vote — Sen. Jeff Merkley, Oregon Democrat, was the lone “no” vote — and now heads to the full Senate, where Majority Leader Mitch McConnell, Kentucky Republican, said he plans a vote next week before Congress’ Independence Day recess.

If the bill clears the Senate, which appears likely given Wednesday’s committee vote, then the House must act. The No. 2 Democrat in the lower chamber signaled that the hefty bipartisan support for the Senate’s bill could clear hurdles on his end.

“We don’t want to leave here without humanitarian resources to handle what is a humanitarian crisis at the border,” said House Majority Leader Steny H. Hoyer of Maryland.

Immigrant rights groups blessed the compromise as a good step, which could help cut down on objections by Democrats who are sensitive to the complaints of the immigrant and Hispanic communities.

One reason for the support among Democrats was their ability to delete money they didn’t like — such as additional detention beds at U.S. Immigration and Customs Enforcement — and to insert restrictions they did want, such as guaranteeing members of Congress access to facilities and imposing new standards on detention and searches of migrants at the border.

Democrats also won language imposing a sanctuary policy on HHS, preventing it from sharing information with Homeland Security about illegal immigrants encountered during the search for sponsors to foster the children.

In most cases, the foster families are the children’s parents or other relatives who are in the U.S. illegally. They often are living in households with other illegal immigrants who sometimes are targets for deportation at ICE. The bill gags federal health officials from reporting that to ICE.

Republicans, meanwhile, emerged with a bill slightly bigger than the $4.5 billion Mr. Trump requested.

The Republicans’ biggest win was symbolic. The bill is an acknowledgment by Congress of the border crisis Mr. Trump warned is happening.

The scope of the crisis is revealed in the numbers.

In May alone, the Border Patrol nabbed nearly 133,000 people trying to sneak into the U.S. Officers at the southwestern land border crossings encountered more than 11,000 others who showed up and demanded entry without permission.

Those aren’t records themselves, but the number of people coming across as part of a family — nearly 89,000 — did shatter the previous record.

Roughly 12,000 more children traveling alone also arrived at the border.

For the families and the unaccompanied children, Homeland Security’s goal has been to process and release them as quickly as possible — directly into communities in the case of the families and to health officials in the case of the unaccompanied children.

Yet they’re coming in such massive numbers that the government simply can’t process them quickly enough.

HHS has 13,700 children already in its care and fewer than 450 additional beds open — but 1,400 are being held by Homeland Security at the border while they wait for a transfer.

Homeland Security says its border facilities are built to hold perhaps 4,000 people, but it has 15,000 in those facilities right now. Last week, the number was even higher at 19,000.

Mr. Merkley, the lone senator to oppose the funding bill in committee, cast some of the blame for the backlogs on Mr. Trump. He said the administration’s get-tough policies are scaring away sponsors who would otherwise host the children.

He said he wanted the funding bill go further in banning some of the types of shelters that HHS has contracted to hold the children.

“Without these changes, President Trump and his administration will use increased funding to lock up more children and inflict suffering on thousands,” Mr. Merkley said.

The funding bill may be the lone bipartisan sweet spot on immigration this year, dealing with care for those already in the U.S.

Far more controversial is trying to figure out how to stop the flow of people making the journey.

Of the 75,000 people who came as families in 2017, some 95% are still in the U.S., ICE says.

Republicans say unless they can find a way to change those numbers, more will come, certain that they, too, can gain quick release and an illegal foothold in the country.

Democrats counter that they want more money pumped into Central America. They say the solution lies not in tighter U.S. enforcement but in nation-building in the chief migrant countries of El Salvador, Guatemala and Honduras.

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