•Tuned inerter damper is proposed to control the vibrations of adjacent bridge structures.•The control effectiveness of the proposed damper is compared with the conventional viscous dampers.•Tuned inerter damper can achieve the same control effectiveness with smaller damping coefficient than viscous dampers.•Vibration characteristic differences between adjacent bridges influence the control effectiveness.•Increasing the absolute stiffness of a bridge improves the control efficiency. This paper proposes using a tuned inerter damper (TID) system to mitigate the potential pounding and unseating damages between the adjacent bridge structures under severe earthquakes. The control effectiveness of the proposed method is investigated both in the frequency and time domains. For comparison, the models and responses of the adjacent bridge structures without control and controlled by the traditional viscous dampers that are modelled by a Kelvin system or a Maxwell system are also developed and calculated. The analytical results reveal that the TID system can achieve almost the same or even better control effectiveness compared to the conventional viscous dampers with a much smaller additional damping. The proposed method can provide a good alternative to control the excessive relative motions between the adjacent bridge structures.