My mother was small all Her life. As was suggested, you need to use the BOM chart for your weight and size. Being to small is not good for anyone. You are young, you are going to want to have a family and being small will not be productive to that. Get out live a little. Mind the charts for age and grow healthy.
No!!! I am 16 as well and though I don't know you, I know what you're going through (to a certain extent) I'm 5'9" and I weigh 180. Technically, I'm only partially overweight, according to the BMI chart. Anyway, I've had friends your height who weighed 90 pounds, and they get sick very easily, and when they get sick, it's bad. Your body will start to shut down because it cannot support itself. The average weight for your height is at least 110. At the very least. Remember that you are wonderfully made, and that you are beautiful! 90 pounds is sick, not healthy, and healthy is beautiful.
Your goal will put you in a category of being far too underweight from the healthy range for a female your age.
Why would you want to be only 93 lbs? (I'm making the assumption that you are presently weighing more, hopefully over 117 lbs)
The concern here is that you might have a serious issue deeper issue which leads to you having a distorted view of a healthy and desirable body image.
The consequences of being severely underweight can be detrimental to your health and well-being.
I suspect that your body fat percentage is already low to sustain healthy hormonal,cognitive and neurological function.
Also, your body might be in a catabolic state, where muscle is used for energy, resulting in muscle loss-same as in muscle wasting!
Very underweight females are prone to anemia, hair loss, amenorrhea,
pregnancy complications, osteoporosis, infections and weak immunity mainly due to low intake and absorption of vital nutrients.
I would start by talking to your physical education teacher, your trainer, your coach, a dietitian, a nutritionist ... to help you formulate a plan to achieve a
better balance between training, nutrition, recovery, etc.
You may need to consult with a therapist in the event that you don't "see" your very low weight goal being a serious issue.