furze, gorse
蒺 refers to thorny, shrubby plants: furze and gorse. The core idea is a prickly, low-growing plant.
蒺 combines the plant radical 艹 with 疾, which likely provides the sound. The character is used for thorny plants, but the exact historical development is uncertain.
The grass radical 艹 sits above 疾, which looks like a sickness under a roof. Imagine a prickly gorse bush growing so fast it looks sickly, its thorns spreading like a rash.
For シツ, picture a gorse bush with thorns so sharp they could puncture a sheet: sheet -> シーツ, but the bush is so dense you can only see the シツ part. The prickly plant shreds the sheet.