High-order epistasis has been observed in many genotype-phenotype maps. These multi-way interactions could have profound implications for evolution and may be useful for dissecting complex traits. Previous analyses have assumed a linear genotype-phenotype map, and then applied a linear high-order epistasis model to dissect epistasis. The assumption of linearity has not been tested in most of these data sets. Using simulations, we demonstrate that neglecting nonlinearity leads to spurious high-order epistasis. We find we can account for this nonlinearity in simulated maps using a power transform. We then measure and account for nonlinearity in experimental maps for which high-order epistasis has been previously reported. When applied to seven experimental genotype-phenotype maps, we find that five of the seven exhibited nonlinearity. Correcting for this nonlinearity had a large effect on the magnitudes and signs of the estimated high-order epistatic coefficients, but only a minor effect on additive and pairwise epistatic coefficients. Even after accounting for nonlinearity, we found statistically significant fourth-order epistasis in every map studied. One map even exhibited fifth-order epistasis. The contributions of high-order epistasis to the total variation in the map ranged from 2.2% to 31.0%, with an average across maps of 12.7%. Our work describes a simple method to account for nonlinearity in binary genotype-phenotype maps. Further, it provides strong evidence for extensive high-order epistasis, even after nonlinearity is taken into account.