In this work, a displacement-amplified torsional damper (DATD) is proposed for improving the seismic capacity of the beam-column joints of a frame structure. The proposed DATD uses common steel, lead, and high-damping rubber. This damper exhibits good energy dissipation under small earthquakes. Under strong earthquakes and large displacements, the strengthening of the high-damping rubber can improve the overall stiffness of the damper and increase the energy dissipation. In order to investigate the performance of the proposed DATD, theoretical analyses, simulations, and cyclic loading tests were performed, and their results were compared, showing an overall good agreement.