  In this work we show the strong convergence of propagation of chaos for the
particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions
as well as for the moderate interaction particle systems on the level of
particle trajectories. One of the main obstacles is to establish the strong
well-posedness of the SDEs for particle systems with singular interaction. To
this end, we extend the results on strong well-posedness of Krylov and
R\"ockner \cite{Kr-Ro} to the case of mixed $L^p$-drifts, where the heat kernel
estimates play a crucial role. Moreover, when the interaction kernel is bounded
measurable, we also obtain the optimal rate of strong convergence, which is
partially based on Jabin and Wang's entropy method \cite{JW16} and Zvonkin's
transformation.
