The phrase "normal weight" refers to a healthy or average weight range for a person based on their height and age. It means being neither too heavy nor too light but staying within a healthy weight range. Full definition
After an initial period of normal weight maintaining diet which for determining their daily calorie requirements, they were divided into three groups. (fitnessandpower.com)
Not too concerned, but it supports your evidence that it likely doesn't work for normal weight range. (paleoforwomen.com)
Their brain metabolism rates were comparable to the activity seen in normal weight women. (sciencedaily.com)