An investigation of two nonlinear mixed Volterra–Fredholm integral equations of the second kind is undertaken. The equations are constructed as fixed-point problems, under the condition of Lipschitz continuity, the Generalized Banach fixed-point theorem is used to prove the existence of a unique solution without any contraction assumption. A double numerical quadrature algorithm is constructed for the occurring two-dimensional integral. It is theoretically proved, under the existence of relevant partial derivatives, that the proposed numerical scheme has (i) first order of convergence if the nonlinear kernel is multi-variate, and (ii) second order of convergence if the kernel is uni-variate. A Newton–Raphson scheme is then designed and implemented for the resulting nonlinear algebraic system. Some numerical examples are used to demonstrate the accuracy and convergence of the method. The numerical results verify the theoretical results and show that the method performs well.