This is nothing to worry about - it sounds like mycelium strands, which, whilst they are fungal in origin, are not only not harmful, but beneficial to plants. They are usually only visible in areas where the soil is very dry - when the soil is moist, they are not noticeable at all when you dig. Given that's the case, the cause of your plants dying isn't the mycelium, its because its too dry at times for newer plants to survive, particularly if you have large cherry laurels nearby. If you want to plant close to the laurels, I'm afraid you need to keep the new plants constantly well watered, or, more sensibly, plant further out/away from the laurels.

A tip I discovered when planting in a very dry area is to add some moisture retaining granules in the planting hole, together with some good compost if you have any. It really does cut down the amount of watering you have to do. And do research the needs of the plants before you buy them - some will hate it in that position whatever you do. Not much will grow under big cherry laurels. One or two ferns will tolerate dry shade - Dryopteris felis mas and Polysticum setiferum are two I've used. If you are short of ideas look at Long Acre Plants website.