Front-end vibration

Car is a 1998 Toyota Camry. At speeds above 60, front end shakes violently, with steering wheel oscillating between 10 and 2 o’clock positions. Seems very dangerous. Recent attempts to fix have been: four brand new balanced Michelins and a four-wheel alignment. This has not helped. Oddly, after braking, problem usually disappears until the next time I drive. Could this be a rotor issue or something to do with rack and pinion?