In this paper, we propose a spectrum sharing scheme relying on non-orthogonal multiple access (NOMA) and cognitive radio (CR) in a hybrid satellite-terrestrial network (HSTN) to improve spectrum efficiency. The primary network shares its spectrum resource with the secondary network on the condition of maintaining normal communications by the criterion of tolerable interference level. Moreover, secondary users share the same wireless resource by employing NOMA to on the one hand further enhance the spectrum efficiency, and on the other hand to be beneficial to massive connection design. Afterward, considering Shadow-Rician fading for satellite links, as well as Nakagami-m fading for terrestrial links, we evaluate the outage performance. In this process, mutual interference between the primary and secondary networks is investigated so that the derived expressions may precisely reflect the realistic performance. Finally, numerical simulation results are presented to highlight the performance of our proposed scheme.