We discuss Bragg scattering on both Bose and Fermi gases with strong short-range interactions in the deep inelastic regime of large wave vector transfer q, where the dynamic structure factor is dominated by a resonance near the free-particle energy ℏω=ϵ=ℏ$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$q$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$/2m. Using a systematic short-distance expansion, the structure factor at high momentum is shown to exhibit a nontrivial dependence on frequency characterized by two separate scaling regimes. First, for frequencies that differ from the single-particle energy by terms of order $O$(q) (i.e., small deviations compared to the single-particle energy), the dynamic structure factor is described by the impulse approximation of Hohenberg and Platzman. Second, deviations of order $O$(q$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$) (i.e., of the same order or larger than the single-particle energy) are described by the operator product expansion, with a universal crossover connecting both regimes. The scaling is consistent with the leading asymptotics for a number of sum rules in the large momentum limit. Furthermore, we derive an exact expression for the shift and width of the single-particle peak at large momentum due to interactions, thus extending a result by Beliaev [J. Exp. Theor. Phys. 7, 299 (1958)] for the low-density Bose gas to arbitrary values of the scattering length a. The shift exhibits a maximum around qa≃1, which is connected with a maximum in the static structure factor due to strong short-range correlations. For Bose gases with moderate interaction strengths, the theoretically predicted shift is consistent with the value observed by Papp et al. [Phys. Rev. Lett. 101, 135301 (2008)]. Finally, we develop a diagrammatic theory for the dynamic structure factor which accounts for the correlations beyond Bogoliubov theory. It covers the full range of momenta and frequencies and provides an explicit example for the emergence of asymptotic scaling at large momentum.