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.