"My grandmother is sick."
Translation:Bà của tôi bị bệnh.
The word ốm only means sick/ill in the North, in the South that same word means skinny/meagre/thin. Bệnh (also spelt bịnh in the past) can mean sick/ill/diseased in all dialects. Here's a summary:
ENGLISH --- NORTHERN --- SOUTHERN
skinny/meagre/thin --- gầy --- ốm
sick/ill --- ốm --- bệnh
[NOTE: bệnh can also be spelt bịnh by some but it's considered a non-standard spelling today even though it was common prior to 1975 in South Vietnam. In fact, South Vietnamese may spell it bệnh today but most will pronounce it like bịnh ("bựn") just as they write nhất but most still pronounce it as nhứt ("nhức").]