It sounds like you have a drainage problem......do you mean fungi or mushrooms?
If mushrooms....could the top soil have mushroom spores in it? I've never heard of laying turf on top of an existing lawn..I would think water isn't able to drain very quickly through the old lawn and due to the recent high temperatures, mould and fungi are developing. I would get the turf layers back to it.

I have the odd mushrooms in my lawn and garden from time to time they tend to go away on their own.Do you have this or much more?
Although there is a school of thought that advocates top soil on top of an old lawn and under turfs will take. I doubt your new lawn is making a proper root system If your old lawn needed replacing it makes no sense to me to just cover it and expect the new turf to be better than your old lawn? but you havent given enough information?

It could just be the turf you've used - this problem does occur sometimes with new turf. The spores may well have been present in it and now the fruiting bodies are showing themselves as your mushrooms and toadstools. Hopefully, that's what it is, because eventually the problem will disappear, though it'll take a year or more probably. If it's poor drainage because of lack of proper preparation before laying, you've a much bigger problem on your hands, so I suggest playing the waiting game, unless you're feeling energetic and want to get out there and lift the lot and prepare properly.Be awful to do all that and then find you've still got the problem because its the turf, not the soil underneath, though if the grass is yellowing and the area's waterlogged, that's exactly what you will have to do, but I'd wait for those signs to appear first.