My new iPhone 13 will not connect to to my Wi-Fi. It’s shows connected but it’s not working

Eliminate your router by trying to connect to someone else's wi-fi.

After that you should reach out to Apple Support directly. Depending on your region, you may be able to reach them here: contact Apple Support 

1. Reset network settings (Settings, General, Transfer or reset).
2. Reset the router to factory settings (usually a paper clip in a hole for 10 seconds)
3. Uninstall any VPN or security app

I had the same problem. In addition, my VPN switch was flickering back and forth. I had a VPN app that wasn't set up/logged on or connected--I hadn't set it up on my new phone yet. I removed the app, and my phone immediately connected to wi-fi. I reinstalled the app and connected it and everything is working fine now.

I’ve tried these steps and have also reset my modem and it is still a problem.

Thank you that confirms my number 3.

VPN's are a curse on iPhones, and iOS 15 has its own "Private Relay" for similar additional protection.

Thank you for the fix. It worked.

You're welcome - just sharing the love,