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In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]