Normalization of spatial transcriptomics data is challenging due to spatial association between region-specific library size and biology. We develop SpaNorm, the first spatially-aware normalization method that concurrently models library size effects and the underlying biology, segregates these effects, and thereby removes library size effects without removing biological information. Using 27 tissue samples from 6 datasets spanning 4 technological platforms, SpaNorm outperforms commonly used single-cell normalization approaches while retaining spatial domain information and detecting spatially variable genes. SpaNorm is versatile and works equally well for multicellular and subcellular spatial transcriptomics data with relatively robust performance under different segmentation methods.