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.