late 15c., "strap passing under the jaw of a horse" (used to restrain the animal), from Old French courbe (12c.) "curb on a horse," from Latin curvus, from curvare "to bend" (see curve (v.)). Meaning "enclosed framework" is from 1510s, probably originally with a notion of "curved;" extended to margins of garden beds 1731; to "margin of stone between a sidewalk and road" 1791 (sometimes spelled kerb). Figurative sense of "a check, a restraint" is from 1610s.
1520s, of horses, "to lead to a curb," from curb (n.). Figurative use from 1580s. Related: Curbed; curbing.