The CO₂ doesn’t always lag behind the temp. What happens is some of the warming periods are initiated by something other than CO₂ and the oceans become a source, and in some cases CO₂ initiates the warming trend and the ocean becomes a CO₂ source. The oceans take longer to cool than does the atmosphere and the surface, so they remain a source well after the global temps start dropping. CO₂ itself, because of the source/sink relationship stays high for hundreds of years.

Because the denialists only look at graphs, all they see are the peaks where temps start down and some centuries later CO₂ heads down. However, the peaks tell us nothing about the original initiator of the warming trend.