Is the ghusl (ritual bath) valid if the white discharge was found alone, then found again later, knowing that there was a non-mucous brown discharge at the time of the first ghusl? And if the ghusl was not valid, what must be done after weeks have passed since the incident?

If a woman sees a yellowish discharge (kudrah) and a clear discharge (tuhr) at the same time, and she does not know which one appeared last, then the ruling is that the clear discharge occurred later. This is because the default state is the continuation of menstruation, and the clear discharge is considered to have occurred at the earliest possible time. Consequently, her ritual bath (ghusl) is valid, and nothing further is required of her.