WASHINGTON—A House committee approved far-reaching legislation to curb the market dominance of tech giants, including Alphabet Inc.’s Google and Facebook Inc., but much of the effort faced intensive lobbying by affected firms that slowed the committee’s work and foreshadowed a pitched battle in the Senate.

The centerpiece of the six-bill package, a measure to bar big tech companies from favoring their own products in a range of circumstances on their platforms, was approved early Thursday by a vote of 24 to 20. Known as the American Choice and Innovation Online Act, the legislation would prohibit big platforms from engaging in conduct that advantages their own products or services, or disadvantages other business users, or discriminates among similarly situated business users.

Another measure, passed late Wednesday, requires that the largest internet platforms make it easier for users to transport their data to other platforms and even communicate with users on other platforms. The bill—known as the Augmenting Compatibility and Competition by Enabling Service Switching, or Access, Act—would give the Federal Trade Commission extensive new powers to set individualized standards for the tech giants. It passed, 25-19.

The bills must still pass the full House, where the timetable for bringing them to the floor for final votes remains unclear. The package would affect large tech firms including Google, Facebook, Apple Inc. and Amazon.com Inc.

The debate over the House Judiciary Committee’s legislation started Wednesday morning and continued into Thursday morning, as Republicans—and some Democrats—raised concerns and floated amendments.

Two other less-controversial bills also were adopted, one raising federal fees on corporate merger reviews and another aiding state attorneys general in procedural battles in antitrust court cases. The panel will consider a final bill in the package when it reconvenes Thursday at 11 a.m.

The package was the culmination of a lengthy investigation by a House antitrust subcommittee. It found that the big tech companies have leveraged their dominance to stamp out competition and stifle innovation, adding that Congress should consider forcing them to separate their platforms from other business lines.

Taken together, the bills represent the beginnings of an effort by many in Congress to reinvigorate antitrust enforcement among high-tech companies by updating laws they say have fallen behind. Rep. David Cicilline (D., R.I.) said the unchecked power of the biggest tech companies threatens economic fairness and even American democracy itself.

“At its core, this issue is fundamentally about whether or not we have an economy where businesses fighting for economic survival can actually succeed,” Mr. Cicilline said.

The effort has gained support from the Biden administration, which recently surprised Silicon Valley firms by naming a young progressive critic of big tech, Lina Khan, as chair of the FTC, one of two federal agencies that enforce U.S. antitrust laws. Ms. Khan is a former House antitrust staffer who worked on the big tech investigation. She is expected to refocus the agency’s enforcement efforts on anticompetitive problems.

But the White House suggested further work might be needed on some of the legislation, reflecting potential problems ahead.

“The president is encouraged by the bipartisan work to address problems created by big tech platforms,” a White House official said. “We hope the legislative process continues to move forward on these bipartisan proposals, and we look forward to working with Congress to continue developing these ideas.”

The legislative effort also has run into fierce opposition from many big tech companies and their Washington allies on both sides of the aisle.

Google pushed lawmakers to delay action on the bills pending more debate. “American consumers and small businesses would be shocked at how these bills would break many of their favorite services,” said Mark Isakowitz, vice president of government affairs and public policy for Google. “This would all dramatically undermine U.S. technology leadership, damage the way small businesses connect with consumers and raise serious privacy and security concerns.”

Apple released a report on Wednesday arguing against provisions of the American Choice and Innovation Online Act that would allow users to download apps onto their iPhones without having to use Apple’s App Store. The company said that would harm customers by threatening their privacy and parental controls and potentially exposing users’ data to ransomware attacks.

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Many Republicans also voiced concern that the package was overreaching by handing too much new power to government agencies, while a few tech-friendly Democrats raised concerns that the legislation had not been adequately refined.

Rep. Jim Jordan (R., Ohio), a vocal critic of the legislation, said it represented a worrisome instance of big tech and big government “now marrying up and working together.” He complained that it would give unprecedented power to the FTC to set industrial policy and even impose its own political agenda on the affected companies.

Some lawmakers also suggested that Microsoft Corp. had lobbied to avoid being covered by the legislation, which generally affects only the largest platforms. Mr. Cicilline denied that the bill exempted any company.

A Microsoft spokeswoman said the company didn’t seek changes to the bill to avoid being affected by the legislation.

Other lawmakers questioned the wisdom of regulating only the largest tech platforms.

The battle is likely to intensify in coming weeks. While antitrust legislation remains one of the bigger vulnerabilities in Congress this year for the big tech companies, several of the committee’s more far-reaching bills face uphill fights to become law in their current form.

In the Senate, Amy Klobuchar (D., Minn.) is leading an effort to pass antitrust legislation and has developed a broad package of changes. She has also been focused on developing more proposals along the lines of some of the House measures, particularly the nondiscrimination bill.

She said in a statement: “I look forward to continuing to work with members of the House and Senate to rein in the unfettered power of big tech.”

Write to John D. McKinnon at john.mckinnon@wsj.com