Is 5'4 150 pounds fatter than most women?

Who cares about average? "Most people" are fat. You're overweight. Overweight=fat.

Regardless of what the BMI scale tells you, this is a perfectly healthy weight. The BMI scale was absolutely never accurate, and yet people will still cling to it for some reason.

The BMI scale does not take into account bone density, muscle mass, sex and racial differences, nor overall body composition. It basically takes the ideal body weight of a single person and then tries to apply it to absolutely everyone else. That's why most doctors don't talk about weight with their patients anymore, because they know that their patients are perfectly healthy and the BMI scale is absolutely wrong.

That being said, a person who has chubby thighs, some fat on their stomach and chubby cheeks is a perfectly healthy person. Someone who looks like the 1000lb sisters is unhealthy and needs help.

That's pretty average