The Eneolithic Botai culture of the Central Asian steppes provides the earliest archaeological evidence for horse husbandry, ~5500 years ago, but the exact nature of early horse domestication remains controversial. We generated 42 ancient-horse genomes, including 20 from Botai. Compared to 46 published ancient- and modern-horse genomes, our data indicate that Przewalski"s horses are the feral descendants of horses herded at Botai and not truly wild horses. All domestic horses dated from ~4000 years ago to present only show ~2.7% of Botai-related ancestry. This indicates that a massive genomic turnover underpins the expansion of the horse stock that gave rise to modern domesticates, which coincides with large-scale human population expansions during the Early Bronze Age.