I must agree with sotongeoff, I would speak with the company who laid the turf, it is supposed to be weed free, and I imagine it was not cheap. Mint can be almost as hard to get rid of as the brambles, but if you keep mowing it is will give up eventually, not much besides grass tolerates having its head chopped off regularly. Also your garden will smell lovely each time you cut, mown grass and mint - lovely idea.