Juvenile growth is submaximal in many species, suggesting that a trade-off with juvenile growth must exist. In support of this, recent studies have demonstrated that rapid growth early in life results in decreased physiological performance. Theory clearly shows that for submaximal growth in juveniles to be optimal, the cost of growth must be nonlinear. However, nearly all of the empirical evidence for costs of growth comes from linear comparisons between fast- and slow-growing groups. It is consequently unclear whether any known cost can account for the evolution of submaximal juvenile growth. To test whether the cost of growth exhibits the logically necessary nonlinearity, we measured critical swimming speed (U_crit), the maximum speed sustained in incremental velocity trials, in Atlantic silversides, a species for which the costs and benefits of growth are well studied. To increase our ability to detect a nonlinear relationship between U_crit, a proxy for juvenile fitness, and growth, we manipulated ration levels to produce a broad range of growth rates (0.16 mm/day⁻¹ to 1.20 mm/day⁻¹). Controlling for size and age, we found that U_crit decreased precipitously as growth approached the physiological maximum. Using Akaike's information criterion, we show that swimming performance decreases with the square of growth rate, providing the first demonstration of a nonlinear cost of growth.