When your assistant prompt includes a URL, the assistant will only use that URL, and won't search the web for additional context.
For example, when asked "give me some articles about chairs", the Assistant compiles a list of chair-related articles.
However, if you ask it "give me some articles about this thing: https://en.m.wikipedia.org/wiki/Chair" it will summarize information from the Wikipedia page but won't provide any additional articles.
It should always search the web if it has to, even if specific URLs are provided.