1540s, probably from Old English wice "Applied generally or vaguely to various trees having pliant branches" [OED], from wican "to bend" (from PIE root *weik- (2) "to bend, to wind") + hæsel, used for any bush of the pine family (see hazel (n.)). The North American bush, from which a soothing lotion is made, was so called from 1670s. This is the source of the verb witch in dowsing.
1853 in the literal sense (witch-hunting is from 1630s), from witch (n.) + hunt (n.). The extended sense is attested from 1919, American English, later re-popularized in reaction to Cold War anti-Communism.
Senator [Lee S.] Overman. What do you mean by witch hunt?
Mr. [Raymond] Robins. I mean this, Senator. You are familiar with the old witch-hunt attitude, that when people get frightened at things and see bogies, then they get out witch proclamations, and mob action and all kinds of hysteria takes place. ["Bolshevik Propaganda," U.S. Senate subcommittee hearings, 1919]
"a remedy which is very effective, almost magical;" see silver (adj.) + bullet (n.). The belief in the magical power of silver weapons to conquer foes goes back at least to ancient Greece (as in Delphic Oracle's advice to Philip of Macedon). In Britain, silver bullets as a superstitious countercharm figure in the fictitious Popish Plot (1678).
'Cause champed silver kills stone-dead
Such as are musket-proof 'gainst lead.
[Thomas Ward, from "England's Reformation," 1710]
English folklore beliefs recorded from early 19c. held that a witch could be wounded or revealed (if transformed) only by a wound from a silver bullet. Similar fancies are reported in folk-tales from Ireland and Iceland. The belief in the killing efficacy of silver bullets was transferred to vampires by 1816.