Genome sequence analysis, which analyzes the DNA sequences of organisms, drives advances in many critical medical and biotechnological fields. Given its importance and the exponentially growing volumes of genomic sequence data, there are extensive efforts to accelerate genome sequence analysis. In this work, we demonstrate a major bottleneck that greatly limits and diminishes the benefits of state-of-the-art genome sequence analysis accelerators: the data preparation bottleneck, where genomic sequence data is stored in compressed form and needs to be decompressed and formatted first before an accelerator can operate on it. To mitigate this bottleneck, we propose SAGe, an algorithm-architecture co-design for highly-compressed storage and high-performance access of large-scale genomic sequence data. The key challenge is to improve data preparation performance while maintaining high compression ratios (comparable to genomic-specific compression algorithms) at low hardware cost. We address this challenge by leveraging key properties of genomic datasets to co-design (i) a new (de)compression algorithm, (ii) hardware that decompresses data with lightweight operations and efficient streaming accesses, (iii) storage data layout, and (iv) interface commands to access data. SAGe is highly versatile as it supports datasets from different sequencing technologies and species. Thanks to its lightweight design, SAGe can be seamlessly integrated with a broad range of genome sequence analysis hardware accelerators to mitigate their data preparation bottlenecks. Our results demonstrate that SAGe improves the average end-to-end performance and energy efficiency of two state-of-the-art genome sequence analysis accelerators by 3.0x-32.1x and 13.0x-34.0x, respectively, compared to when the accelerators rely on state-of-the-art decompression tools.