Does the water used to wash a defiled part of the body become impure, specifically when cleansing oneself after urinating (istinja) or performing ghusl, especially if the defiled area is touched while water is flowing over it?

These are reprehensible whispers (waswas), and their cure is to ignore them. When water flows over an impurity and separates from it without being changed by it, many scholars rule that it is pure. Giving in to waswas leads to great evil.