The sliding of isolation bearings in a bridge structure can reduce bridge pier damage, but bearing sliding under the strong earthquake may cause excessive displacement of the girders or even falling girders. This study proposes an innovative shape memory alloy (SMA) cable-restrained sliding lead rubber bearing (SMA-SLRB), which can maximise the sliding advantages of the bearing whilst combining the displacement-limiting capacity and self-centering capacity of the SMA cable. The working mechanism of the SMA-SLRB is firstly revealed, and the mechanical model of the SMA-SLRB is verified by tests. Then, the multilevel fortification performance target of isolated continuous girder bridges with SMA-SLRB is proposed. A displacement-based seismic design (DBSD) method for the SMA-SLRB isolated continuous girder bridge is established. Next, with the single pier model of a five-span isolated continuous girder bridge taken as an example, the displacement-based design is implemented, and the seismic performance of the bridge is evaluated. Finally, the seismic responses of bridges with SMA-SLRB and SLRB under extremely rare earthquakes are compared. Results show that the energy dissipation capacity and stiffness of SMA-SLRB are improved to varying degrees compared with SLRB. The bridge basically achieves the design target displacement and meets the requirements of multilevel fortification performance, which confirms the effectiveness of the DBSD method. Under extremely rare earthquakes, the SMA-SLRB can effectively reduce the maximum displacement and residual displacement of the girder compared with the SLRB.