Is this phrase grammatically correct?

My (internet) connection is not very good but I can watch Youtube videos without them freezing.

É mais comum o uso da palavra 'Youtube' como um adjetivo (descrevendo 'videos') em vez de um possessivo nesta circunstância.

Is this phrase grammatically correct?

My connection is not very good but I can watch Youtube's video without freezing.

In answering to your question, yes, it is grammatically correct and understandable.  However for the older generations, they may not understand the techo-lingo of today.  That's Ok, they just have to learn what Youtube etc is.

Your sentence can be better expressed as,

"... watch Youtube without frezzing."  This is because YouTube is what all videos is about.  So using the word "video" is redundant.

Is this phrase grammatically correct?

My connection is not very good but I can watch videos on YouTube without my computer freezing.

(I think it should be like this, but you did a good job!)