In the present paper, the regularity and finite difference methods for Caputo-Hadamard fractional differential equations with initial value singularity are taken into consideration. To overcome the weak singularity and enhance convergence precision, a fitted scheme on nonuniform meshes is applied to such problems. Firstly, based on Llog,2-1σ approximation, the temporal convergence accuracy of fitted scheme for the sub-diffusion equations is O(N) , where N denotes the number of time steps, α is the fractional order and r is the mesh grading parameter. It is indicated that the performance of the fitted scheme is better than that of the standard Llog,2-1σ scheme on (i) exponent meshes (i.e., r= 1) and (ii) graded meshes with the optimal choice of the mesh grading. Secondly, a second-order fitted scheme on exponent meshes for the diffusion-wave equations is obtained. Furthermore, for the sake of improving the computational efficiency and demonstrating the effectiveness of the decomposition of the solution, the fast algorithm and further decomposition of the solution for the sub-diffusion equations are investigated. Ultimately, some examples are presented to verify the availability of our theoretical results.