That "white stuff" is the keratin, which is what nails are composed of. It is not anything accumulating under your nails.
It is opaque white shows near the nail bed where it keratin is thicker, and lighter (more transparent) near the nail tip has thinned due to scrubbing with a toothbrush and cleaning with a sharp instrument. The attachment of your nail to the nail to skin, also known as the nail bed, is also normal, trying to remove it can damage your nail, or open places for fungus or infection to penetrate.
Here is a detailed description of what your nail is and how it grows:
https://my.clevelandclinic.org/health/body/24734-nail-matrix
If you are still uncertain, you could stop in at your school nurse (if you have one) or a local nail shop to ask if your nails look normal to them.
Sue