The reason for wearing tzitzit, as clearly expressed in the Torah, is that we should see them and be reminded of G‑d's mitzvot, and be protected against the heart's tendency to stray. This goal is accomplished when the tzitzit fringes are actually visible. This is the reason for the custom to have the tzitzit strings dangling from beneath the shirttails.
Tzitzit aren't inherently holy. Therefore, wearing tzitzit when entering a washroom poses no problem. It is customary, however, to remove a tallit before entering a restroom. This is an act of respect accorded to a garment which is reserved for prayer.
