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.