You'd be best burying the turf upside down, at least six inches if you can manage it. If you just turn it over and leave it on top the new turf will have nothing to root into and you could lose the lot. It would also be very difficult to produce a good level surface.

In my professional opinion i would not suggest turning over the old lawn not only does is damage the soil but the existing grass could grow back through, its better to take the time, by removing the turf completely and either seeding the area or re tufing.

You can dig up your turf and put the upside down turfs in some other area of the garden then prepare the soil on the old lawn area before laying new turf. If you already have a lawn though! good maintenance may well do just a good a job as new turf, which will only be as good as the the old one next year if you dont look after it? (if needed you can just resow any bald bits)
Regular mowing and weed/feeding scarifing will transform most lawns in one/two seasons. Keeping the grass short encourages side growth and thickens it up and weakens weeds.

I think Drc has the best idea, try to save the lawn you have is a good start.

Benjamin, please could you explain why burying turfs would damage the soil? I have always understood that when they rot down they improve it. I bury mine quite deep and they never grow through. Of course you have to let the ground settle before you returf.

Good question ST, I always put mine upside down in an area I am trying to improve soil wise as my lawn gets less and less each year. But to dig them in under a new lawn may make it become bumpy as they rot down and sink?

Yes - I did say let it settle, but should have said it takes quite a while! Sorry - perhaps not the best suggestion in this instance. But it does solve the problem of disposing of the turf if you have a large area to cope with and throwing it out seems such an awful waste. Still can't see how it would spoil the soil though.

Rather than spread them out you can also stack them upside down in a tidy pile ST.

Yers...I have done that in the past but never moved the tidy pile afterwards, so now I have two little hills covered with grass etc.!

Must be good stuff inside each pile ST?

You could plant them up and make a feature ;))

Not where they are. And the most accessible one has a little hole in it and I think someone is living there, so I won't disturb him. The other one is now covered with couch grass, ho hum. One of these days...