Is ghusl after fifteen days of continuous yellowish discharge following menstruation valid, and is the white discharge that turns yellowish on cotton considered a sign of purity or not?

The questioner's action is incorrect. The yellowish discharge connected to blood is considered menstruation. If the duration of the blood and the yellowish discharge connected to it exceeds 15 days, then the woman is in a state of istihadah (prolonged vaginal bleeding). She must refer to her previous habit, if she had one. Otherwise, she should differentiate between menstrual blood, based on its characteristics, and istihadah blood. If she has neither a habit nor the ability to differentiate, then her period is six or seven days. After the end of the menstrual period, she should perform ghusl, take precautions (like wearing a pad), and perform ablution for each prayer after its time has begun. Vaginal discharges are not considered menstruation; they are pure but invalidate ablution, unlike the impure yellowish discharge.