My mum reckons that when I grow it will "stretch out", do I have anything to be worried about or not?

I know you might think and feel differently now, but you are in the category of people with normal weight, so maybe your stomach o some other area might seem like too much, the thing is that all you basically have to do is make some corrections in your activities and things you eat. Anything more than that would make you underweight and since you are now hitting puberty, it's absolutely normal that your body and your body image are changing. Your mom is right - time will probably be all it takes for these things to solve themselves.