There are Some steps that you should follow to play music on YouTube with the screen turned off.
Step 1: Enable desktop site in a browser.
Step 2: Install the VPN app and set location to the United States.
Step 3: Open YouTube music.
Step 4: Search and play the music.
Step 5: Minimize the app to play YouTube music in the background.
That's how you can play music on YouTube with the screen turned off.
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