can i lay turf over weed suppressing mat and have it grow.

its possible but defeating the object as the weeds will just grow in the soil with the grass

Not to mention that you'd need some soil on top of the membrane, or mat, for the turf to root into. Why don't you just lay turf properly - it works fairly well at keeping weeds out if you cut it regularly.

and the mower naturaly takes care of the weeds well most of and the rest you can get selective weedkiller.

If it is a small area , just fork out the perennial weeds like Couch and Gout Weed with running roots. The turf should suppress all\ other annual weeds.