There is a complicated grammatical reason, which I will try not to mangle in a second, but the simple answer is that where you have two nouns together like this in a way that could be translated "the x of the y", e.g., "the sea of the north" or "the street of the king", in Welsh you cross out the first "the" and the "of".
More formally, this is a genitive construction in Welsh, and the Welsh genitive is formed by putting two nouns or noun phrases together with the possessor coming second, and only the possessor takes the definite article.
English has two ways of forming the genitive, coming from two different routes. You can say "The cat's mother" or "the mother of the cat". Just remember that in Welsh it is always the second form. Mam y gath = the Mother of the cat. Write out that English form and cross out all but the last "the" and the "of" and you have it.
Of course you might not think that "The North Sea" is a genitive construction at all, but literally in Welsh it is "The Sea of the North". the sea that belongs to the North. So you see it does make sense.
I don't quite agree with it but I do understand from your position that say this. What I mean is quite straight forward - that is "Dw i" - I "ddim" - not / do not "yn" - in / is "hoffi" - like / liking "Mor" - sea "y" - the "Gogledd" - North. Therefore I say "I do not like sea the north" or "north the sea" vice versa. Come to this point, I nearly want to give up, not to continue anymore.... serious. Quite tough
A word for word translation?
Dw (Am) i (I) ddim (not) yn (in) hoffi (liking) Môr (Sea) y (the) Gogledd (North)
Don't worry about feeling frustrated. Celtic languages can be a bit funny if you're new to them. I see you're already an experienced language learner (I'm impressed!), so I'm sure you can get to grips with Welsh too. I'm happy to help if you've got questions.
How about you take a little rest and come back to Welsh in a day or two after you've had a break?