You're 16, 5'1" and 150lb? You're probably going to hate me for this, but you are NOT "fat"!! Another thing you'll hate me for - you're still so young, still growing, how do you know in four years time you won't weigh 100lb without dieting?
I've been on both sides of the coin; from 12 I was desperate to lose my 'puppy fat' and just stopped eating. It worked, obviously. But at a cost - I looked awful, skin and bone, terrible skin, no energy, in pain all the time. By my early 20s and after my first child I was the opposite. 5'4" and almost 20st. Could not stop eating and didn't exercise at all. Agoraphobia didn't help. I then got roped into joining slimming world and it did wonders for me. I was eating healthily for once and never felt better. That's not a 'diet', it's just good eating habits and being healthy mind and body. Now, I'm back not eating again because of anxiety, it's not fun at all. I don't want to become overweight again either but if I could bring myself to adopt that healthy lifestyle again I would.
Don't let this rule your life. If you're honestly that unhappy then join somewhere like I did, just do it safely with expert advice and support. Everyone here saying EDs aren't fun are SO right. You don't want to go down that road.