The Eneolithic Botai culture of the Central Asian steppes provides the earliest archaeological evidence for horse husbandry, similar to 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 similar to 4000 years ago to present only show similar to 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.