Hmm, are you using WiFi or getting Internet over your cellular network connection?? I'm wondering if you have some kind of cellular data plan with your provider that limits video if you've exceeded some usage limit?
Anyway, in no particular order, some things I'd try (one at a time and see if videos work afterwards):
0) Update your iPhone to latest iOS available for your device
1) See if it fails when you're using WiFi and also when you're using Internet over your cellular connection
2) Close and reopen Safari (https://support.apple.com/en-us/HT201330)
3) Power off WiFi and turn it back on
4) Put the iPhone into Airplane mode and take it back out
5) Power off/on the iPhone (not just lock the screen, fully power off and on)
6) Take it to an Apple retail store and have them check it
Good luck 
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