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Bacteria play an indispensable role in marine biogeochemistry by recycling dissolved organic matter. Motile species can exploit small, ephemeral solute patches through chemotaxis and thereby gain a fitness advantage over nonmotile competitors. This competition occurs in a turbulent environment, yet turbulence is generally considered inconsequential for bacterial uptake. In contrast, we show that turbulence affects uptake by stirring nutrient patches into networks of thin filaments that motile bacteria can readily exploit. We find that chemotactic motility is subject to a trade-off between the uptake benefit due to chemotaxis and the cost of locomotion, resulting in an optimal swimming speed. A second trade-off results from the competing effects of stirring and mixing and leads to the prediction that chemotaxis is optimally favored at intermediate turbulence intensities.