A bad tie rod's end, ball joint or even bearing could be the cause of your vibration when turning the wheel. Use your jack to raise the front wheels. Strongly shake each front wheel to feel any play. The way you described it the play must be obvious when shaking the wheel. If there is a play then do the same thing with the tie rod to see if the play comes from the tie rod's end.
If there is no play and if your car is a front wheel drive then the problem could come from the universal joints (also called cardan). A worn out cardan can create vibration when turning the wheels. A worn out cardan can be identified by a ' clok, clok' sound when turning the wheels to the maximum while the car is moving.