When I was 16-17, I was a chronic pot smoker, cig smoker, and all around piece of shit. I weighed 155 lbs and was 5ft10. I was skinny, I didn't eat 3 meals a day, I slept 4-5 hours a day, and I rarely exercised. I too have small wrists and hands.
I grew to 6ft by the time I was 21. I still weighed 160lbs. I started lifting at 21 and now 7 years later I'm 194lbs. I bench over 300lbs. I still consider my wrists to be small, but it hasn't affected my strength. I naturally have a thin frame so when I pack on muscle it is way more noticable. Small wrists make your biceps, triceps, and forearms look fucking huge.
My point is don't stress out about your wrists because it won't affect your strength. Eat 5 meals a day with a decent source of protein in each meal. Lift at least 3 times a week and always add weight. You'll be a fucking beast in no time.
I'm telling you this because I wish I had someone to tell me this when I was your age.