DHAKA, Bangladesh—The trade war between the U.S. and China has led many fashion brands to shift production to spots across Asia, including to Bangladesh, where safety issues persist years after two workplace accidents killed more than 1,000 workers.

Many American fashion brands cut back on sourcing from Bangladesh, and others abandoned the country entirely, after the 2012 and 2013 accidents. Over the past two years, however, companies have bulked up sourcing in the country amid efforts to improve safety conditions there, and...