I'm growing fatter, but my wieght remains the same. Is that normal?

Normal, not healthy. The most likely explanation is that you are replacing lean muscle with fat. The only solution to this is exercise which improves your health in so many ways that I defy anyone to identify a disease whose incidence and/or prognosis isn't improved with regular exercise. Ideally exercise during which you have sustained heavy breathing for a minimum of 20 minutes, three times/week.