In this work, we obtain quantitative convergence of moderately interacting particle systems towards solutions of nonlinear Fokker-Planck equations with singular kernels. In addition, we prove the well-posedness for the McKean-Vlasov SDEs involving these singular kernels and the trajectorial propagation of chaos for the associated moderately interacting particle systems. Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. This seems to be the first time that such quantitative convergence results are obtained in Lebesgue and Sobolev norms for the aforementioned kernels. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the Sobolev regularity of infinite-dimensional stochastic convolution integrals, and we also exploit the regularity of the solutions of the limiting equation.