In this paper, we consider the Galerkin method to approximate the solution of Fredholm–Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O(n−r), whereas the iterated Legendre Galerkin method converges with the order O(n−r−α+[Formula presented]) for the algebraic kernel, and order O(lognn−r−[Formula presented]) for logarithmic kernel in both L2-norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence O((1+clogn)n−r−2α+[Formula presented]) for the algebraic kernel, and order of convergence O((logn)2(1+clogn)n−r−[Formula presented]) for logarithmic kernel in both L2-norm and infinity norm. Numerical examples are given to illustrate the theoretical results. © 2018 Elsevier B.V.