The paper is concerned with $L^p$ error analysis of semi-discrete Galerkin FEMs for nonlinear parabolic equations. The classical energy approach relies heavily on the strong regularity assumption of the diffusion coefficient, which may not be satisfied in many physical applications. Here we focus our attention on a general nonlinear parabolic equation (or system) in a convex polygon or polyhedron with a nonlinear and Lipschitz continuous diffusion coefficient. We first establish the discrete maximal $L^p$-regularity for a linear parabolic equation with time-dependent diffusion coefficients in $L^∞(0,T;W^{1,N+\epsilon}) \cap C(\overline{\Omega} \times [0,T])$ for some $\epsilon>0$, where $N$ denotes the dimension of the domain, while previous analyses were restricted to the problem with certain stronger regularity assumption. With the proved discrete maximal $L^p$-regularity, we then establish an optimal $L^p$ error estimate and an almost optimal $L^∞$ error estimate of the finite element solution for the nonlinear parabolic equation.