Dynamic routing and congestion control are two major problems in software-defined hybrid satellite-terrestrial multicast networks research. Due to terrestrial users being allowed to join or leave the multicast group at any time and the differences between the satellite and the terrestrial networks, many multicast routing algorithms reroute rapidly and thus increase the rerouting overheads. Meanwhile, the congestion ratio is increased by some hot nodes of satellite-terrestrial link transmission paths. This paper focuses on rerouting overheads and congestion problems in satellite-terrestrial multicast networks. We present a satellite-terrestrial network architecture with the Software-Defined Networking (SDN) features to offer dynamic multicast services for terrestrial users. A Two-Layered Shared Tree Multicast (TSTM) routing algorithm is proposed to achieve efficient dynamic multicast group management, address the trade-off between bandwidth consumption and rerouting overheads. The algorithm also implements congestion control by using a load factor to reflect on the global network bandwidth usage in routing calculations. This algorithm balances the rerouting frequencies of satellite and terrestrial networks to decrease the rerouting overheads and also reduces the network congestion ratio. The simulation shows TSTM decreases rerouting cost, user time delay, and node congestion ratio compared with the locality-aware multicast approach (LAMA).