Relating to, resembling, or having a strong interest in horses; or looking like a horse (often used to describe someone's facial features).
From Old English 'hors' (horse) with the suffix '-y' added to make an adjective meaning 'having the quality of.' The word has been used since at least the 1600s.
The word 'horsey' became especially loaded in British high society, where it described wealthy aristocrats obsessed with horse racing and horseback riding—it's actually a pretty coded social marker of a certain type of privilege and lifestyle.
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