Viruses from downloading videos

Correct. As above, A popup is a bogus advertising message, hoping to trick you into downloading some garbage app, or worse still, contacting some criminal, masquerading as a “support site”. Do note, the more often you engage with sites that are offering “free” content, the more likely you are to see more of these messages.

Phony "tech support" / "ransomware" popup… - Apple Community

No, it is not necessary. Safari and iOS are blocking most trackers by default. But if you want to be extra careful, do your downloads in a Private tab in Safari. Then any history or website data from that tab will be discarded when you close it.

See Turn Private Browsing on or off on your iPhone - Apple Support

Note that in general, the kinds of web sites that have scary pop-ups are the kind that help you do things that you are quite possibly inadvisable or illegal. Downloading videos from YouTube is against their terms of service. Although I used to do that too.

While it is not possible to infect your phone by doing so, do understand that if you are downloading any internet content for “free” that is normally not available for free from the rights holder, you will see no end of bogus advertising posted. The criminals posting the bogus messages are generally the same as those providing “free” content, that is normally not available for free.

You are welcome.