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Dynamic graph learning is crucial for accurately modeling complex systems by integrating topological structure and temporal information within graphs. While memory-based methods are commonly used and excel at capturing short-range temporal correlations, they struggle with modeling long-range dependencies, harmonizing long-range and short-range correlations, and integrating structural information effectively. To address these challenges, we present SALoM: Structure Aware Temporal Graph Networks with Long-Short Memory Updater. SALoM features a memory module that addresses gradient vanishing and information forgetting, enabling the capture of long-term dependencies across various time scales. Additionally, SALoM utilizes a long-short memory updater (LSMU) to dynamically balance long-range and short-range temporal correlations, preventing over-generalization. By integrating co-occurrence encoding and LSMU through information bottleneck-based fusion, SALoM effectively captures both the structural and temporal information within graphs. Experimental results across various graph datasets demonstrate SALoM's superior performance, achieving state-of-the-art results in dynamic graph link prediction. Our code is openly accessible at https://github.com/wave5418/SALoM.