Don't let go the handlebars! Some bikes yes, some bikes no, some tyres yes, some tyres no. You could spend a lot of money trying to cure a problem that's no big deal, just don't let go the bars!
This can come from the front or back, weight pressing on the headstock can amplify wobble from the back.
I couldn't tell you if mine shakes its head or not, don't remember letting go the bars! TKC's probably cuse vibration so I'll stick to keeping a hold of the bars!