What to do when "Weak Security" message shows by Wi-Fi user name?

It tells you what to do. Your router is set to use WPA and WPA2. WEP (the oldest Wi-Fi security) was broken long ago. WPA replaced it and that too is no longer secure. iOS is letting you know your router is broadcasting an easily hacked wireless security protocol, and you should turn WPA (not WPA2) off.

From your Mac or Windows computer, go into your router's web setting pages. You should be able to do this from your iPhone, too, as long as you're within Wi-Fi range of your router. Typically, you put 192.168.0.1 into the URL search field. The router should prompt you for an admin name and password. If you don't know what those are, almost all newer routers have that info on a sticker, which is on the router itself.

Once in the settings pages, go to the wireless settings. You should be able to find a drop down menu for the security options. Change it to use WPA2 only, or WPA2/WPA3 if you have that option. Save the settings (the router may tell you it has to be restarted).

One possible downside. If you have much older devices that connect to the router that don't understand any security protocol newer than WPA, they'll no longer be able to talk to the router.

Hi - this is an issue of the security setting on your router or wifi access point, and is nothing to do with your phone.

The message, which appears under the Wi-Fi network name on your iPhone, states that WPA/WPA2 TKIP is “not considered secure”

You need to change the setting on your router so it is at WPA2 AES or better.