As it starts at about 60mph and without more specific detail it sounds like an alignment problem, either that or there could be damage to one of the wheel rims which is forcing the wheel to resist the pressure against the road when turning instead of giving to it which a tyre is supposed to do.
I see you say that you have had the wheels balanced , but have you had the wheels electronically aligned by a competent tradesman?
Have you checked out the whole suspension / damping system?
Is the car making any noises?