Searching for similar genomic sequences is an essential and fundamental step in biomedical research and an overwhelming majority of genomic analyses. State-of-the-art computational methods performing such comparisons fail to cope with the exponential growth of genomic sequencing data. We introduce the concept of sparsified genomics where we systematically exclude a large number of bases from genomic sequences and enable much faster and more memory-efficient processing of the sparsified, shorter genomic sequences, while providing similar or even higher accuracy compared to processing non-sparsified sequences. Sparsified genomics provides significant benefits to many genomic analyses and has broad applicability. We show that sparsifying genomic sequences greatly accelerates the state-of-the-art read mapper (minimap2) by 2.57-5.38x, 1.13-2.78x, and 3.52-6.28x using real Illumina, HiFi, and ONT reads, respectively, while providing up to 2.1x smaller memory footprint, 2x smaller index size, and more truly detected small and structural variations compared to minimap2. Sparsifying genomic sequences makes containment search through very large genomes and large databases 72.7-75.88x faster and 723.3x more storage-efficient than searching through non-sparsified genomic sequences (with CMash and KMC3). Sparsifying genomic sequences enables robust microbiome discovery by providing 54.15-61.88x faster and 720x more storage-efficient taxonomic profiling of metagenomic samples over the state-of-the-art tool (Metalign). We design and open-source a framework called Genome-on-Diet as an example tool for sparsified genomics, which can be freely downloaded from https://github.com/CMU-SAFARI/Genome-on-Diet.