Dunelm have a good selection of fitted sheets in different depths. I buy a 32cm depth for my thick memory foam mattress and once it's stretched over the mattress there's no need to iron it because the creases get smoothed out in the process.
LilyoftheValley, maybe your fitted sheet is too big, because if it fits correctly it's quite tight, and isn't going anywhere! On the other hand, if it's too small and the corners don't tuck under the mattress enough, it will ping off!!
Maybe sodapop has the right idea and flat sheets would be easier 🤷.