Fungi are under nail. With a fungal infection of a nail, such as toenail fungus, the fungi live underneath the nail. Bleach can kill germs, including fungi, on one's toes and on the outside surface of the toenails. However, the fungi under the nails will likely survive and keep growing.
No. The only way to "cure" a nail fungus is by taking oral medicine, which has recently been found to be much safer than we originally thought. Also: not all thick, discolored nails are fungal. The only way to tell is with a fungal culture of the nail.