Apple TV not working
How the picture will come on TV from you tube
iPhone 11, iOS 15
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Do you have the Apple TV box? Identify your Apple TV model - Apple Support
With the current models (4th generation or later), you would be able to download the YouTube app from the on-device App Store. The YouTube organization has stopped supporting the older models Apple TV (3rd generation or earlier), except for AirPlay.
The TV app on your iPhone can’t help you with YouTube.
The Apple TV app on your smart TV or streaming device can’t help you with YouTube.
To stream YouTube content from the YouTube app on your iPhone to a TV set, you may need to have AirPlay or Chromecast functionality on the receiving TV. Use the casting symbol to start streaming to other devices. It may require a setup procedure the first time. Note that this is an in-app feature of the YouTube app on iPhone. It may work differently for other apps (AirPlay but no Chromecast).