The secondary lamellae (thin, leaf-like protrusions from the gill filament) are best described as a sheet of blood separated by an array of posts (the pillar cells) which keep the lamellum from ballooning, maintaining it one red blood cell thick.
A resting teleost fish typically removes 80% of the oxygen from the water passing over it's gills. Some experimental results for the tench showed a mean efficiency of 51% oxygen removal. Reversal of the water flow reduced the efficiency to 9%. A typical mammal, such as a human removes only 10 to 20 % of the oxygen from air. Why such a contrast in efficiency? Consider the medium. Water is 800 times as dense as air. Air normally contains 20% oxygen (200,000 ppm). Water, by contrast contains only about 10 ppm oxygen and under stagnant conditions this may drop much lower. Fish have to be more efficient.
If fish gills are so efficient and air contains so much more oxygen, why do fish suffocate out of water? Surface tension of water clinging to the gill collapses the lamellae and the effective surface area is reduced to a tiny fraction of normal. Catfish survive better than most other fish because they have cartilage supports for the primary filaments and have the secondary lamellae thickened and separated more than usual.
Hemoglobin exhibiting the Bohr effect will require a higher oxygen tension to fully load under conditions of low pH. Hemoglobin showing the Root effect can never be fully saturated with oxygen when pH is low.