How can one get rid of purity-related intrusive thoughts (waswas), especially concerning the emission of madhy (pre-seminal fluid) and wady (thick, white discharge after urination), the cleanliness of clothes and place, and the water reaching the ankles or covering the head during ablution (wudu)?

The optimal cure for obsessive doubts (waswasah) is to ignore them and pay them no heed. If they return, cast them away and focus on your worship. The default assumption is that ablution (wudu) is not invalidated and that nothing has exited unless there is absolute certainty. The default assumption for things is purity. So, do not scrutinize and overthink, for that is a form of extremism (tanattu') and surrender to obsessive doubts. The predominance of belief is sufficient for the completeness of wudu and ghusl (ritual bath). Responding to obsessive doubts opens the door to evil and brings hardship, and Allah Almighty desires ease for us.