"guide the course of a vehicle," originally by a rudder or something like it, Old English steran (Mercian), stieran (West Saxon) "steer, guide, direct; govern, rule; restrain, correct, punish," from Proto-Germanic *steurjanan (source also of Old Norse styra, Old Frisian stiora, Dutch sturen, Old High German stiuren, German steuern "to steer," Gothic stiurjan "to establish, assert"), related to *steuro "a rudder, a steering," from PIE *steu-ro- (source also of Greek stauros "stake, pole"), extended form of root *sta- "to stand, make or be firm."
The notion is of a stiff, upright pillar or post used in steering, or else perhaps "establish," hence "direct, steer." Intransitive sense also was in Old English. To steer clear of in the figurative sense of "to avoid completely" is recorded from 1723. Related: Steered; steering.
"young ox," Old English steor "bullock," from Proto-Germanic *steuraz (source also of Old Saxon stior, Old Norse stjorr, Swedish tjur, Danish tyr, Middle Dutch, Dutch, German stier, Gothic stiur "bull"), perhaps from PIE *steu-ro-, denoting "larger domestic animal" (see Taurus). In U.S. of male beef cattle of any age.