We propose a biquadratic characteristic finite volume element (CFVE) method for two classes of time-fractional convection-dominated equations on quadrilateral meshes. For the mixed time-fractional problem (i.e., involving both timefractional and integer-order derivatives), optimal convergence rate in $L^2$-norm is established for the CFVE method with temporal discretization based on the $L1$-formula. For the pure time-fractional problem, we decompose the equation by separating an integer-order time derivative term, thereby enabling the application of the method of characteristics. Furthermore, the influence of the decomposition parameter is investigated through numerical experiments, providing practical insights into parameter selection. Extensive numerical experiments validate the efficacy of the proposed CFVE method in simulating both classes of time-fractional convectiondominated equations.