Theories in which gravity is weaker on cosmological scales have been proposed to explain the observed acceleration of the universe. The nonlinear regime in such theories is not well studied, though it is likely that observational tests of structure formation will lie in this regime. A class of alternative gravity theories may be approximated by modifying Poisson’s equation. We have run N-body simulations of a set of such models to study the nonlinear clustering of matter on 1–100 Mpc scales. We find that nonlinear gravity enhances the deviations of the power spectrum of these models from standard gravity. This occurs due to mode coupling, so that models with an excess or deficit of large-scale power (at k < 0.2 Mpc-1) lead to deviations in the power spectrum at smaller scales as well (up to k ~ 1 Mpc-1), even though the linear spectra match very closely on the smaller scales. This makes it easier to distinguish such models from general relativity using the three-dimensional power spectrum probed by galaxy surveys and the weak lensing power spectrum. If the potential for light deflection is modified in the same way as the potential that affects the dark matter, then weak lensing constrains deviations from gravity even more strongly. Our simulations show that, even with a modified potential, gravitational evolution is approximately universal. Based on this, the Peacock-Dodds approach can be adapted to get an analytical fit for the nonlinear power spectra of alternative gravity models, though the recent Smith et al. formula is less successful. Our conclusions extend to models with modifications of gravity on scales of 1–20 Mpc. We also use a way of measuring projected power spectra from simulations that lowers the sample variance, so that fewer realizations are needed to reach a desired level of accuracy.