This paper presents a study on the energy-absorbing capacities of viscoelastic dampers (VEDs) for reducing the resonant vibrations of simply supported high-speed railway bridges of short to medium span. The proposed solution is based on retrofitting the bridge with a set of discrete VEDs connected to the slab and to an auxiliary structure, placed underneath the bridge deck and resting on the abutments. In this investigation attention is focused on mitigating flexural vibrations; therefore, both the bridge and the auxiliary structure are modelled as simply supported beams with Bernoulli–Euler (B–E) behavior, whereas a discrete fractional derivative model simulates the behavior of the damping material. Firstly, a parametric study of this planar model is carried out, which has led to a dimensioning procedure of the dissipative system. The technical feasibility of this particular retrofit design is numerically evaluated by applying it to a numerical model of a simply supported railway bridge with inadmissible vertical accelerations. Numerical results show that the dynamic response of the structure can be significantly reduced in resonance with the proposed damping system