The finite-difference discretization of the Riesz space-fractional diffusion equations under consideration yields a series of linear systems whose coefficient matrices consist of the identity matrix and the product of diagonal and Toeplitz matrices. By leveraging the Toeplitz structure contained in the discrete linear systems, we design a banded preconditioner with shift compensation (BSC) to improve the convergence rates of Krylov subspace iteration methods. The BSC preconditioner can be regarded as a modification of the existing banded preconditioner with diagonal compensation (BDC). Unlike the BDC preconditioner, however, it preserves a Toeplitz-like structure similar to that of the coefficient matrix of each discrete linear system. Theoretical analysis and numerical experiments demonstrate that the BSC preconditioner can outperform the BDC preconditioner or at least achieve comparable performance.