Hi Huu
I have now put three raised beds over grass, filled with organic horse manure six inches deep, planted veg straight off and everything has grown superbly. No cardboard or any other covering was used and no weeds or grass resurfaced. A bed I put in over soil gave me a minor problem with Japanese anemone coming up in a couple of places for a year or two but this has finally given up the ghost (she said with fingers crossed!).
I’ve also skimmed off the top inch or so of turf, put down a porous membrane and put gravel on top of a small area near the house where cold frames are and did the same preparation but placed bark on top of another area. All worked well for me but these were small areas only.