Maybe you have the same issue discussed in this topic:
https://forums.opera.com/topic/2584/opera-won-t-load-youtube

thanks a lot dude x) u fixed my problem! Apparently i just had to turn off Off-Road mode (donno what it means/what it is for but it worked) thanks a lot

well, i just tried it is not working (thanks though). What is wierd, is that YouTube is working on google chrome and IE just not on opera

u got the same problem?

Hi guys! Lately i got some problems with youtube.. For some reasons, Opera cannot open any youtube page. When i try to launch Youtube, I just see a white blank page.And if i go to a different website that use the youtube plugin, well the youtube plugin just doesn't appear.
Anyone know what causes this or how to fix it?