I find it ridiculous that people are diagnosing it as a problem with the shock or the wishbone for definite without having seen the car or listened to the noise.
Shock absorber failure usually results in a knocking noise over bumps, and it would have to have one of the mounts physically broken for it to rub on the tyre. The fact that the noise only happens when the car is moving tells us its something to do with a rotating component, either the driveshaft or the tyre rubbing on something. There would have to be something really wrong with the steering setup for the tyre to rub on the wishbone, but the plastic of the inner arch can easily be dislodged and come close enough to the tyre for it to rub on it at full lock, hence my earlier diagnosis, but as I said, it's almost impossible to say without the OP having a look inside the wheel arch to see if anything's wrong.