Recently, low-orbit satellite networks have gained lots of attention from the society due to their wide coverage, low transmission latency, and storage and computing capacity. Providing seamless connectivity to users in different areas is envisioned as a promising solution, especially in remote areas and for marine communication. However, when jointly used with terrestrial networks composing satellite-terrestrial networks, the satellite moving speed is much faster than the ground terminal, which can cause inconsistent service from a single satellite, and therefore lead to frequent satellite handover. Moreover, due to the dynamic and time slot visibility of satellites, the topology of an intersatellite changes frequently, which results in loops during satellite handover, thereby reducing the utilization of links. To address these problems, we propose a digital twin-assisted storage strategy for satellite-terrestrial networks (INTERLINK), which leverages the digital twins (DTs) to map the satellite networks to virtual space for better communication. Specifically, we first propose a satellite storage-oriented handover scheme to minimize the handover frequency by considering the limited access time and capacity constraints of satellites. Then, a multiobjective optimization problem is formulated to obtain the optimal satellite by genetic algorithm. Finally, considering the timing visibility of the satellite links, a digital twin-assisted intersatellite routing scheme is introduced to improve the quality of data delivery between satellites. Simulation results demonstrate that the proposed INTERLINK can reduce both handover times and average propagation delay compared with its counterparts. Meanwhile, benefitting from integrated DT, both the quality of data delivery and the delay of intersatellite links are considerably improved.