I wrote this whole long response about pressing seams, but when I looked again at the 2nd picture I saw something else. I could be way off but ......
The first and second rows of 4patches look fine (at the top of the 2nd pic) - the individual blocks lined up with nesting seam allowances, and it looks like a nice long flat seam when you joined the rows and pressed to one side.
So that should just carry on to the rest of the rows.
However, if you look at the last row of 4patches (the one closest to the camera), those blocks are turned a different way - so the long seam on the 4 patch is now horizontal instead of vertical (per the picture's orientation).
You might not have realized when you were sewing and pressing all those 4 patches together that they weren't all being assembled in the same manner. Of course, from the front they all look the same, but not the back, and that's where the seams are giving you a problem on this last row.
What do to about it?
You could just sew this seam and then press it open. This would distribute and eliminate some of the bulk. You just have to make sure the seam is properly opened on the back, flat on the front, and there are no tucks or pleats.
The other option is to take that last row apart, and repress it so it's just like the other blocks. But that would be a royal PITA. Because it wouldn't just be changing the orientation of the blocks - it would be repressing each one, and unless you were going to take it back to the 2patch sub-unit to really correct the issue, it's might not be worth the effort.
btw, a twisted seam here and there on the back is not the end of the world. It happens. As long as it's properly pressed on the front - with no tucks or pleats and a flat seam, it won't cause a problem. Well, maybe if you're hand quilting.
And I want to add that I think your checkerboard background is fabulous. You say you're a new quilter, but it's great that you're not just doing the same old, same old. It is a lot a more work than just plain patches, but it really catches the eye. Worth it!
I just located my first quilt top from 1998/9-ish. It was also a TATW. The seam allowances (of greatly varying widths) were pressed in very interesting directions within the same seam, and there was nary an intersection that matched up properly. I got a very good chuckle out of it. ;-)
Everyone has to start somewhere.