You're not doing anything wrong. This is common; some babies are like this. There are possible reasons, but there's also unexplained crying. It's called colic.
My first was very colicky. It started when she was 3 weeks old. One night she cried for almost 8 hours straight.
If you want to try some things to see if you can make it better, you can try eliminating some foods from your diet that might bother her. Some common ones are spicy foods, veggies like broccoli or cauliflower that can cause gas, and dairy. None of those things helped my dd.
Also, sometimes that kind of crying can be caused by reflux, although my understanding is the baby would be more likely to cry at or after every feeding if it was reflux. I'm not sure - I don't know a whole lot about it and dd's doctor didn't think that was her problem.
Chances are, it's just colic (unexplained baby crying). "Just" is both reassuring and at the same time makes you want to say, "There's no JUST about it!!"
To cope with it, dh and I would switch dd back every 10 minutes while she cried. It was very stressful for us and lasted about 2 months. The thing with the switching back and forth is that you agree that while your partner is holding the baby, you get to do something else and try not to focus on or worry about the baby. When the 10 minutes is up, your turn again to try to make the baby feel better, or at least hold and comfort while crying. Our other rule was that if the baby stopped crying while you were holding her, then you kept her until she started crying again.
Lots of motion is one thing my dd liked. I spent a lot of time walking up and down the hall with her in the carrier (she like the Bjorn), or rocking my body, or giving her a little gentle up-and-down jiggle. Dh used to sit on the edge of the bed and bounce up and down and sing.
It was really hard to get through those months.
It WILL pass, though. You aren't doing anything wrong. He should be much better by 3 mos old. I have only heard from one parent who said colic for her baby lasted much beyond 3 mos.