How to make half of a square circular I have a square. I want half of it to be circular, like this

(image is from a topdown view) 

I kept loop cutting but I couldn't find a way to make it truly "round" and do it evenly.

NOTE: to be very specific, it should only be round horizontally, the rest should be untouched. Like this: 


(please excuse my horrifying drawing)
Add a Cylinder ShiftA > Mesh > Cylinder, set the Cap Fill Type to Nothing.
Select half of it and delete with X or Delete 
Select the four vertices along the edge
Extrude to make the long sides

Fill the back, top and bottom to get this.



Note the top and bottom are Ngons, witch in many cases are undesirable. To get a quads all over the top use grid fill.