I am sorry to say, but based on your photos these are etchings in the paint and appear to be rather severe.
This is likely a result of hard water or bird droppings being left on the paint for an extended period of time. Both of these types of contaminants can cause severe damage such as this, sometimes to the point that the only solution is to have the effected panel repainted.
In your case, I do not expect machine compounding or polishing to make much, if any, improvement, but it is worth trying nevertheless.
If, as I suspect, the process of machine compounding and polishing does not yield any drastic improvement, your only other option before heading to the body shop would be to attempt to wetsand the areas to remove the etching, then compound and polish the area to remove the sanding marks. This is a rather delicate process and should not be done unless you are quite confident in what you are doing.
In the end, these etchings may have completely penetrated the clear coat which means that nothing you do will remove them 100%, and the only option would be to have the area(s) repainted.
Let me know if you've got any other questions
-Zach