Is the fast considered valid if the yellow discharge is considered a sign of purity from the twelfth day of menstruation, despite its continued presence in large quantities, or must those days be made up?

If the yellowish discharge is continuous with blood, then it is menses and purification (Tuhr) does not occur until it stops. It is obligatory to ascertain purity for ghusl (ritual bath), prayer, and fasting. If you are unsure about the time of purity, it is estimated to occur at the earliest likely time, and what precedes it is considered menses. So, if you did not ascertain purity until the fifteenth day, then all the days before that are considered menses, and fasting on those days is not valid and must be made up.