If you've got both the YouTube and Chrome for iOS apps installed, they can work nicely together and allow you to open the links you tap on in the YouTube app in the Chrome app instead of Safari.
If you prefer Chrome over Safari, this setting is one you'll definitely want to use. Here's how to set it up:
That's all there is to it. Links you tap on within the YouTube for iOS app will now launch in the Chrome app instead of Safari. Note that you must have Chrome installed. If you try to enable this feature without having Chrome already installed on your iPhone or iPad, you will be routed to the App Store to download it instead.