Non-terrestrial networks (NTNs) using flying objects such as satellites play key roles in the next-generation wireless system (6 G). The NTN links with long propagation delay and random packet losses pose a great challenge to the performance of Transport Control Protocol (TCP), which many Internet applications rely on. Performance enhancing proxy (PEP) is an easy-to-deploy approach for improving TCP's performance. In this paper, we design and implement a novel PEP called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PEPesc</i> which has two distinctive features. First, it features <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">retransmission-free</i> loss recovery, using an adaptive packet-level forward erasure correction method called streaming coding (SC). Second, as packet losses are recovered by SC, the congestion control problem is simplified to rate control and local acknowledgement between entities based on bandwidth estimation. Based on a queueing theoretic analysis of the design, we carefully devise a protocol and implement PEPesc as an open-source application. Extensive evaluations show that PEPesc can achieve much higher <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">and</i> smoother goodput than the canonical TCP variants and than other existing open-source PEPs in applications including <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iperf</i> and HTTP-based adaptive streaming, and achieves similar performance in web browsing. Finally, we also present a deployment case over a real-world geostationary satellite link.