The Polish team of Kamil Sieradzki, Jakub Majerski, Ksawery Masiuk and Kacper Stokowski touched in at 3:04.46 in Tuesday's final, 0.81 seconds behind second-place Italy.
Team USA won the gold, setting a world record of 3:01.66, Polish state news agency PAP reported.
The freestyle relay bronze marks Poland's first medal at the 2024 World Short-Course Swimming Championships, which began in Budapest on Tuesday and run until Sunday.
Poland won two medals, a silver and a bronze, at the 2022 World Short-Course Swimming Championships in Melbourne, Australia.
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Source: PAP