Is money earned from advertisements that appear on a website considered unlawful (haram), despite filtering out Sharia-non-compliant advertisements, given the possibility that a non-compliant advertisement might appear for a few hours before being discovered and removed?

Your agreement with an intermediary to advertise on your website falls under the category of ijarah (leasing/renting), as you are leasing your website's page for the purpose of publishing advertisements. It is stipulated that the contract must be for advertisements permissible by Sharia. If the intermediary places a non-compliant advertisement, you must remove or prevent it. The payment you receive is permissible because the original contract was for a permissible benefit (publishing advertisements under the aforementioned condition). Generally, for ijarah to be valid, both the rent and the duration must be known to both parties.