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Abstract In this paper, a fast multiscale Galerkin algorithm is developed for solving the boundary value problem of the fractional Bagley–Torvik equation. For this purpose, we employ multiscale orthogonal functions having vanishing moments as the basis of the trial space, and we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which leads to a fast algorithm. We show the algorithm has nearly linear computational complexity (up to a logarithmic factor). Numerical experiments are presented to illustrate the efficiency, accuracy and convergence of the proposed algorithm. Also, comparisons with some other existing methods are made to confirm the reliability of the algorithm