New versions of HTTP protocols have been developed to overcome many of the limitations of the original HTTP/1.1 protocol and its underlying transport mechanism over TCP. In this paper, we investigated the performance of modern Internet protocols such as HTTP/2 over TCP and HTTP/3 over QUIC in high-latency satellite links. The goal was to uncover the interaction of the new features of HTTP such as parallel streams and optimized security handshake with modern congestion control algorithms such as CUBIC and BBR over high-latency links. An experimental satellite network emulation testbed was developed for the evaluation. The study analyzed several user-level web performance metrics such as average page load time, First Contentful Paint and Largest Contentful Paint. The results indicate an overhead problem with HTTP/3 that becomes more significant when using a loss-based congestion control algorithm such as CUBIC which is widely used on the Internet. Also, the results highlight the significance of the web page structure and how objects are distributed in it. Among the various Internet protocols evaluated, the results show that HTTP/3 over QUIC will perform better by an average of 35% than HTTP/2 over TCP in satellites links specifically with a more aggressive congestion algorithm such as BBR. This can be attributed to the non-blocking stream multiplexing feature of QUIC and the reduced TLS handshake of HTTP/3.