In this paper, a compact finite difference scheme with O(τ+h) convergence order for nonlinear Caputo–Hadamard fractional sub-differential equations is proposed, where τ represents the maximum step size in temporal direction, h represents the step size in spatial direction, and α is the order and r (r≥1) is an optional constant. First, we derive the implicit solution of the original equation using the modified Laplace transform and the finite Fourier sine transform. To obtain the regularity, an auxiliary function t is applied to handle the nonlinear term, which is crucial to the analysis. Second, we approximate the Caputo–Hadamard fractional derivative with the L formula on non-uniform grids. Furthermore, we adopt the Newton linearized method to handle the nonlinear term carefully. Based on the discrete fractional Gro¨nwall inequality, the stability and convergence of the derived scheme are obtained by the energy method. Ultimately, three examples are presented to show the effectiveness of our method.