The instructions apply to all Fire tablet models including theFire HD Kids Edition.
Can You Block YouTube From a Kindle Fire Tablet?
How Do You Stop YouTube Videos From Appearing on a Fire Tablet?

Tap theParental Controlstoggle switch.
Enter a password and selectFinish.
This password will be required to adjust parental controls in the future, so don’t forget it.

TapAmazon Content and Apps.
The text beside it will be changed from Unblocked toBlocked.
Scroll down and tapWeb Browserto change it from Unblocked toBlocked.

TapBackto return to the previous screen, then scroll down and tapPassword Protection.
This will require the password you set up to download any app (including the YouTube app).
To hide the Amazon app store altogether, tapAmazon Storesto change it from Unblocked toBlocked.
To remove or set further restrictions, go back to theParental Controls>Apps & Games.
To block YouTube for all users on your Wi-Fi internet, you canblock websites with your router.
It should stop appearing when you pop pop the app or website.
TheYouTube Kids appprovides more options for parents to block individual users and channels.
The easiest way to stop seeing ads is with a subscription toYouTube Premium.