We study the performance exhibited by the transport protocols, Transport Control Protocol (TCP) and QUIC, over realistic satellite networks. We propose a novel methodology, which combines real implementation (exploiting virtualization techniques) and simulation, to carry out systematic and repetitive experiments. We modify the default operation of the ns-3 framework and we integrate the dynamism that characterizes satellite communication links, particularly Low Earth Orbit (LEO). We carry out a thorough assessment over different setups, changing the operating frequency band and packet buffer lengths. In addition, we ascertain the impact of using the multi-streaming feature that QUIC integrates. The results show that QUIC yields lower delays than TCP, although it might suffer from higher jitter in particular setups. In addition, the results evince that using multiple streams in QUIC does not yield a relevant gain for the default Round-Robin (RR) scheduler. We propose more appropriate scheduling strategies, which are able to yield better performances with unbalanced traffic. Even if the behavior of transport protocols over non-terrestrial-networks might not be always appropriate, the obtained results evince that QUIC can definitively bring benefits when compared to TCP. Furthermore, we have shown that optimal scheduling policies yields a fairer performance when using multiple flows, having unbalanced traffic loads.