This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations. We firstly establish a three-level finite difference scheme for the time variable followed by the linearized technique of the nonlinear term. Then the fourth-order compact finite difference method is employed to discretize the spatial variables. Hence the accuracy of the discretization is $\mathcal{O}(\tau^2+h_1^4+h_2^4)$ in $L_2$-norm, where $\tau$ is the temporal step-size, both $h_1$ and $h_2$ denote spatial mesh sizes in $x$- and $y$- directions, respectively. The rigorous theoretical analysis, including the uniqueness, the almost unconditional stability, and the convergence, is studied via the energy argument. Practically, the discretized system holds the block Toeplitz structure. Therefore, the coefficient Toeplitz-like matrix only requires $\mathcal{O} \big( M_{1}M_{2} \big)$ memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O} \big( M_{1}M_{2} (\log M_{1}+\log M_{2})\big)$ computational complexity by the fast Fourier transformation, where $M_1$ and $M_2$ denote the numbers of the spatial grids in two different directions. In order to solve the resulting Toeplitz-like system quickly, an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate. Numerical results are given to demonstrate the well performance of the proposed method.