DHAKA, Bangladesh—The armies of people who make clothes for Western brands—some of the lowest-paid factory workers in the world—are protesting for better wages, a fresh sign that the era of ultracheap labor, and ultracheap clothes, on which many companies rely is increasingly under strain.

Garment workers in Bangladesh make as little as $3 a day, or about $75 a month. In recent days, tens of thousands have refused to work, calling for the minimum wage to be raised to nearly three times that amount. Demonstrations have spiraled, with factories set ablaze and machines smashed. Some three hundred factories were forced to stop operations.

The core of the protests, that workers need higher wages to make even a basic living, has drawn widespread support, including from fashion giants H&M, Gap and Zara-parent Inditex, which source from the country. But no one can agree on who should foot the bill.

Factory owners in Bangladesh say that for them to hike wages in a big way, Western brands that are their top buyers need to pay more for the clothes they order. Although the big names in fashion publicly support higher pay, in practice they balk when costs go up and threaten to shift their orders to other countries, said Faruque Hassan, president of the Bangladesh Garment Manufacturers and Exporters Association.

In a late-September letter, Hassan urged the American Apparel & Footwear Association, an industry body, to persuade clothing brands and retailers to raise prices for clothing orders. “This is important for a smoother transition to a new wage scale,” he wrote, saying that factories were in a “nightmare situation,” facing weak global demand.

Part of the problem is that international clothing brands are also under pressure and grappling with stiff competition, said Rubana Huq, chairperson of Mohammadi Group, a Bangladeshi conglomerate that supplies many of them. “Every time we ask for even 1 cent [more] from the buyer, it is very difficult to get their support,” she said.

Bangladesh’s minimum-wage board is holding negotiations involving both labor and industry representatives to settle on a new minimum wage for garment workers. Factory owners say that if workers’ demands for a roughly $205 minimum monthly wageare met, Bangladesh would lose its competitive edge. Their proposal for a $95 minimum wage was dismissed as unworkable by the government.

Western brands say they support a boost in the minimum wage, though most haven’t said by how much. Workers sewing clothes for them often make substantially more than the minimum wage but far less than what unions are demanding and what international organizations that benchmark living-wage standards consider to be enough.

The American Apparel & Footwear Association said its members are committed to responsible purchasing practices.

Hamim Shikder, a sewing operator who joined protests, says the roughly $125 a month he earns—after 50 hours of overtime—is spent on his son’s school fees, as well as increasingly expensive groceries and other essentials. When his wife recently got sick with dengue he had to borrow money for medical bills. “Whatever amount I get, I have to spend,” he said.

An H&M spokesman said the company supports a new minimum wage to cover the living costs of workers and their families. He declined to say if H&M would pay higher prices to facilitate higher wages, but pointed to a document about its purchasing practices that shows it has a mechanism to allow improved wages to be reflected in the price offered to factories.

An Inditex spokeswoman pointed to recent public statements it made about its commitments to support a living wage for workers in its supply chain. Gap didn’t respond to a request for comment.

The fast fashion model of Western brands is rooted in the premise of low Asian wages. Brands compete to deliver clothing as cheaply as possible, squeezing factories to accept low prices. Factories in turn squeeze workers on wages. These industrywide challenges are magnified in Bangladesh, which is the world’s lowest-cost major garment producer.

“Because of the intense market competition, factory owners in Bangladesh must also strike a balance between raising the minimum wage and maintaining competitiveness,” said Sheng Lu, an associate professor of fashion and apparel studies at the University of Delaware.

Data provided by H&M shows that last year the roughly 600,000 Bangladeshi workers in H&M’s supply chain earned an average of $134 a month. That is above the minimum wage but less than half the $293 a month that Cambodian workers making its clothes earned. Cambodia is poorer per capita than Bangladesh but has a substantially higher minimum wage for garment workers, close to the levels Bangladeshis are demanding.

H&M workers in India earned 10% higher wages than their Bangladeshi counterparts, according to the data. H&M sources much more clothing from Bangladesh than it does from India or Cambodia.

German shoe and apparel maker Puma said in its 2022 annual report that the wages its suppliers paid to Bangladeshi workers were well above the minimum wage, but were only 70% of what a third-party organization had assessed to be a living wage. In some other Puma production centers, such as Cambodia and Vietnam, average pay exceeded the living-wage benchmark.

In a statement, Puma said it was important to approach wage matters collectively, as the challenge cannot be addressed by a single brand. The company said many key suppliers in Bangladesh had policies to ensure worker incomes cover family needs, but said the company still has “a lot to focus on” to turn its policies into further action.

Bangladesh’s garment industry employs millions of people, mainly women, and has been instrumental in reducing poverty. But it also has a history of poor treatment of workers. In 2013, after a factory collapse killed more than 1,100 people, international brands and factories began changes to improve conditions.

Workers’ advocates generally acknowledge that the industry is now safer, but say wages are still too low. Recent calls by Western brands for higher pay ring hollow, they say.

“They are trying to insulate themselves reputationally from a sourcing model that is deeply inhumane,” said Scott Nova, executive director of the Worker Rights Consortium, a labor-rights organization based in Washington, D.C.

Mosammat Champa Khatun, who works in a garment factory and is protesting for higher pay, says she opted for inexpensive Islamic schooling for her child. Even so, her earnings of $110 a month go toward rising transport, food, and housing costs.

“I cannot save,” she said.

Write to Jon Emont at jonathan.emont@wsj.com