Karen, perhaps it is more what you are adding to your heaps as you build them, as opposed to weed seeds blowing in at the top?
After you have removed a few leaves for a salad you can probably get a good quality hoe in between the plants and bosh those weeds on a warm sunny day.
How’s your latest heap doing Rhys? I’ve just built one without any water; comfrey, straw, fresh manure, grass and weed clippings from some verges….fingers crossed!