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This paper introduces a numerical approach by generalizing Legendre wavelets for solving nonlinear Caputo–Hadamard fractional differential equations. The methodology involves the extension of classical Legendre wavelets, namely the generalized Legendre wavelets (gLWs), along with the development of operational matrices for Hadamard fractional integration and Caputo–Hadamard fractional differentiation. The proposed method combines the gLWs with the Adomian decomposition method to address the nonlinearities inherent in fractional equations through Adomian polynomials. A detailed methodology is presented for applying the proposed method to nonlinear Caputo–Hadamard fractional differential equations, accompanied by error analysis and numerical simulations to demonstrate its reliability and accuracy.