Washing a soiled carpet

Q: I am having trouble learning how to purify a carpet. I honestly feel like I have transferred filth in so many places, it must be likely that my salah are not counting. What is the difference between a thick and thin carpet?

I put water three times on impure cat vomit while drying (not completely) with towels in between. But the carpet is thin enough that some water gets to the bottom if I saturate it with enough water. Did I do this incorrectly and now filth is transferred everywhere since later on peoples feet touched that wet area?