Since the App Store won't serve the modern app to your device, you must download a legacy IPA (iOS App Store Package) from a trusted archive.
A reliable source for version-specific IPAs. You can find collections like All YouTube IPAs which include versions compatible with older 32-bit and 64-bit architectures.
This is the most stable way to install an IPA without a jailbreak.
However, you can still get YouTube running by using an or specific legacy workarounds. Below is a comprehensive guide on how to download, install, and fix the YouTube app for iOS 10.3.4. 1. Where to Download the YouTube IPA for iOS 10.3.4
Installing an IPA on iOS 10.3.4 usually requires a computer or a specialized sideloading tool. Method A: Using AltStore or Sideloadly (Recommended)