I'm not sure exactly what you mean by sharpening on the two sides an equal number of times. Do you sharpen on the right 20 strokes then do 20 strokes on the left, or do you alternate 1 right, 1 left, 1 right, 1 left...
You can remove material the fastest by working on one side at a time, but this is for your rough work. When I'm working on a very dull knife I may do 50 to 100 strokes on one side then switch and do the same number on the other side (with my coursest hone). I do this until I feel a burr on the opposite side of the blade from the side I'm honing. I try and get a burr along the whole length of the edge.
Now what you need to do is get rid of the burr and finish off the edge. The burr is made of thin metal like aluminum foil. It is very ductile (bends easily like foil) and does not wear off real easily. Up to this point it is really acceptable to stroke the blade any old direction on the stone, you don't have to stroke edge first when you are roughing down a blade. When you get to the time to get rid of the burr you need to stroke edge-first as if you are shaving the hone.
When you are removing the burr it helps to have a clean hone (with the surface not clogged with metal particals and powdered stone). I like to use a medium fine hone and if it is clogged I scour it with water and sink cleanser. Starting on the burred side hone edge-first until the burr seems gone from that side (maybe 10 strokes). Don't stroke real hard and you may want to hone at a slightly higher angle than your rough work. You are now relatively even on the two sides of your edge. Alternate stroking on each side (left, right, left, right...) using medium pressure for 10 or 20 cycles. Move on to your finest hone and repeat. Do it again using light pressure.
To get your final edge do a few strokes edge-last to pull up any remaining burr. Then use some very light alternating strokes edge-first. Finally tip your edge up to a slightly higher angle and do 5 or 10 cycles of alternating strokes with almost no pressure at all.
Strop your blade on leather (an old belt will do). It should be razor sharp.