It is just inherent in the TN panel itself. There is no way to get rid of it. This is why (among several other reasons) that a lot of people praise the IPS technology. The viewing angle is a lot greater. The technology is a lot better. Even most TV's have better panel technology than these junk TN panels. But, no, there is no way to get rid of it. 
 
What you are seeing, is by turning the monitor sideways it exacerbates the issue. Normally, you would notice the horrible viewing angle more when standing up and looking at it from the very top of the monitor (or the far bottom). Now that you turned it in portrait mode, it shows up by moving your head side to side (which is significantly worse). 
 
EDIT: BTW, there are several 21-24 inch IPS monitors that are sold just below 300 bucks. These would work great for what you are trying to do.