Search the web
Let the assistant look things up, and check what it actually read.
By default the assistant answers from what it knows. The Search button lets it go and look — useful for anything recent, anything specific to a named site, and anything you want citations for.
Using it
Tap Search above the input, then ask your question. While it runs you see what it is doing: Searching the web…, then Reading: example.com as it opens individual pages.

When it finishes, the reply carries a Sources used block. Expand it with Show more sources — the control names the count, so it reads Show 5 more sources when there are five — to see everything it consulted, not just what it quoted.
Check the sources. The value of search is that the answer is traceable. If an answer matters, open the source rather than trusting the summary of it.
Pasted links
You do not need the Search button to give the assistant a specific URL. Paste the link into your message and it will fetch the page itself — AODex used to have a separate “fetch this URL” control and retired it once that became reliable.
When research gets cut short
Long research runs are bounded. If the assistant hits the limit you will see Research was cut short — answered from what was gathered, and the reply is built from the sources it managed to read rather than from the full set it intended to.
That is a real caveat, not a cosmetic one: treat the answer as partial, and either narrow the question or ask it to continue on the specific thread you care about.
Availability
A persona can switch web search off, in which case the button is not shown.