This paper presents a cross-layer design of error-control protocols with rate adaptation for free-space optical (FSO) burst transmission in satellite communication systems. While the prior works mostly focused on the design of error-control protocols for terrestrial-based FSO communications, using either hybrid automatic repeat request (HARQ) based stop-and-wait mechanism or pure sliding window ARQ, we instead propose a design of HARQ-based sliding window mechanism for the high data-rate and long-distance of FSO-based satellite systems. In particular, we model and analyze the performance of low-earth orbit (LEO) satellite-to-ground FSO systems over atmospheric turbulence channels when truncated incremental redundancy (IR) HARQ protocols, which combine the rate-compatible punctured convolutional (RCPC) codes and sliding window ARQ, are employed. For this purpose, the time-varying behavior of atmospheric turbulence channels is first captured by a finite-state Markov chain. Then, the channel model is used to develop the burst loss model to analytically derive the system performance metrics, including throughput, energy efficiency, and average frame delay. The results quantitatively demonstrate the impact of atmospheric turbulence and pointing error on the system performance and support the optimal selection of parameters. Additionally, the effectiveness of the proposed system is numerically confirmed. Monte Carlo simulations are also performed to validate the accuracy of theoretical derivations.