Debunking the myth that the geological record proves CO2 is a consequence of warming, not a cause.

 
"CO2 increase follows temperature increase"

In the geological record that is true, and well understood.
The change in CO2 was not the trigger in those cases.  Rather, some other known event triggered the initial increase, such as large-scale volcanism.

But those triggers alone do not explain the eventual temperature rise that occurred.  The initial rise caused an increase in CO2 (for well understood reasons, such as CO2 release from soils and seas) which then magnified it.  This feedback resulted in the eventual rise being far more than it would otherwise have been, and is one of the reasons what we are doing now is so scary.  Worse, we are adding CO2 into this system which was not then available for release.

(You might wonder why this does not go on forever, leading to more and more GHGs and higher and higher temperatures.  The feedback is not a simple straight-line relationship.  Eventually, extra GHG has less effect.  At the same time, the hotter Earth radiates more heat into space, leading to a new equilibrium.)

However, the fact that the initial trigger was a temperature change on those occasions does not explain the extent of the lag, and neither does it explain that temperature changes precede CO2 changes at other parts of the cycle.

The recorded temperatures and CO2 are both at the Antarctic.  While the CO2 can be expected to represent the global state, the temperatures may be more local.  Andrey Ganopolski and D.M. Roche show that temperature changes tend to be greatest in the Antarctic first, whereas CO2 levels are driven more by temperatures elsewhere.  Hence the data show temperature changes preceding CO2 changes.