You don't have to throw them out ... but perhaps you should combine them with some "grown up" pieces to help you look more like a grown up. I'm 57 and I still wear jeans, shorts, t-shirts, etc. -- I just don't wear them to work.
But yes, if you are wanting to be taking seriously as a professional, you should stop presenting yourself as a teenager. But that can usually be done with just a few minor changes and not a whole, dramatic and expensive overhaul. Buy a few pairs of nicer slacks that aren't skin tight, maybe a skirt or two, a couple of tops that don't advertise products or musical groups, etc. Make sure that when you are dressed, you're underwear doesn't show and you are showing neither too much skin or cleavage. Go for clean and neat, with just a touch of sophistication -- and you'll be fine. Choose basic colors so that you can mix and match to get more for your money. Wear shoes that both look and feel comfortable.
Do you wear make-up? If you don't, maybe it is time to start. You don't need a lot -- and do not need to look all "made up and fake." But maybe a little mascara or blush or lipstick would help. Maybe a new hairstyle, etc. Nothing dramatic - just not something that shouts that you are a teenager.