Are these creeping thistle with a long white root or annual milk (sow) thistle?
In either case I would not put them on to your compost heap. The former will grow again from bits of root and/or seeds. Either let it dry and burn, stuff it into plastic bags and leave for a year, or send to the council recycling where the compost making achieves temperatures high enough to kill roots and seed. The latter will grow from seed. I would send it to the council recycling as if you try to dry it for burning the seeds will scatter.
Creeping thistle is quite vunerable to glyphosate herbicides(e.g. Roundup). My approach would be to spray, wait 2 weeks, dig out the roots. No need to wait for them to die off.
