you have a 7 week old baby, and he is looking at porn!
I agree with jno, that he does not seem to have his mind on fatherhood.
After the euphoria of having the baby, your hormones afterwards are all over the place, tiredness and the demands of a small thing which you never would imagine could make you feel so ragged.
Your partner sounds very immature, which is not helping you. Some men get jealous when the baby arrives, as all the attention is no longer on them, but focussed on you and the baby. He may be reverting to this childish behaviour to get attention. Where he should be focusing his attention on you and your child.
It reads to me, that its his behavior that is causing you to feel like this. You have done a wonderful thing.
You have done nothing wrong. But if you still feel low, have a talk with your health visitor
I wish you lots of luck, as have been there, and know how it feels xx