In this paper, for quickly solving one- and two-dimensional space-fractional sine-Gordon equations with distributed delay, we suggest several accelerated schemes of direct compact difference (DCD) methods. For one-dimensional (1D) problems, with a function transformation, we construct an indirect compact difference (ICD) method, which requires less calculation cost than the corresponding DCD method, and prove under the appropriate conditions that ICD method has second-order (resp. forth-order) calculation accuracy in time (resp. space). By extending the argument for 1D case, we further obtain an ICD method for solving two-dimensional (2D) problems and derive the similar convergence result. For ICD and DCD methods of 2D problems, we also give their alternative direction implicit (ADI) schemes. Moreover, for the fast implementations of ICD method of 1D problems and indirect ADI method of 2D problems, we further present their acceleration strategies. Finally, with a series of numerical experiments, the findings in this paper are further confirmed.