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Hi NewsonyuserC, welcome to the community!
If the phone is only a few days/weeks old, I do highly advise taking it back to the retailer to see if they are able to provide any assistance as this issue has occurred so early on in the handsets life.
Something you can try before taking it back or if they weren't able to assist - is clearing the data for both Youtube and the app you are using to view your home made videos. Just go into Settings > Apps > All > scroll down to the app & tap it > then tap Clear data. If the issue persists after doing so, I then advise booting the phone into Safe Mode to see whether or not the issue is present then, as this will allow us to determine whether it is just a third party app or a software issue.
To boot your phone in safe mode press the power key until the Power Off message appears > hold your finger on Power Off for 2-3 seconds until 'Reboot to safe mode?' appears > tap OK. (To exit Safe Mode just power the phone off & then on normally). If the issue was not present in Safe Mode, that indicates a third party app is causing the issue. In this case exit Safe Mode and then un-install any third party apps you don't need and clear the cache/data for the apps you want to keep. If this doesn't help or you have found that the issue was still present in Safe Mode - you will need to run a software repair on the handset.
Do you encounter the same youtube issue while on >> safe mode ?
as for the MP4 videos, how did you transferred them to your phone? Media Go?