In this paper, fast compact finite difference schemes are derived for fourth-order multi-term fractional sub-diffusion equations with initial singularity under the first Dirichlet boundary conditions. In contrast to the direct scheme, the fast algorithm adopted to approximate the Caputo derivative reduces the computation costs effectively. Sharp error estimate of the proposed scheme for linear models is rigorously presented by the energy method. Furthermore, to handle more intricate nonlinear models, a crucial Grönwall inequality is then deduced to analyse the stability and convergence of the linearized scheme. It is worth pointing out that the Grönwall inequality is also helpful in numerical analysis of multi-step schemes for other problems, such as, integro-differential equations with multiple fractional derivatives. Ultimately, numerical examples are provided to verify the efficiency of the established difference schemes.