It is not obvious that "the GH gas CO2 can also cool the atmosphere" (Singer, American Thinker, April 2, 2018).  In general, this may happen in a region where the lapse rate is positive — i.e., where temperature increases with altitude, as it happens in the stratosphere.

This theoretical prediction is empirically confirmed by Antarctic observations by Schmitthuesen et al. (GRL 2018).  The IR emission from the very cold surface has separated from the emission of the warmer stratosphere.  This supposition can be checked by looking at AIRS (Fourier Transform) data.

This cooling by CO2 will reduce somewhat the normal greenhouse warming by CO2, which will increase roughly as the logarithm of the CO2 level in the atmosphere.  But we must also take into account the at least partial amplification by the (uncertain positive) water vapor (W.V.) feedback (IPCC).

But any CO2 plus W.V. effects will be overshadowed by climate oscillations, like PDO (Pacific Decadal Oscillations) and by changes in solar activity that affect the intensity of cosmic rays (Laster, Lenchek, and Singer in JGR) and the Svensmark cosmic-ray-atmospheric-cloudiness-temperature hypothesis.

The forcing of CO2 depends on its atmospheric level.  One can plot a graph that shows a linear rise at first, until about 70-80ppm.  The curve then becomes logarithmic, so for all practical purposes, beyond 100ppm, the dependence is logarithmic (see Myhre et al. in GRL).

CO2 is a powerful absorber of 15-micron radiation.

But during glaciations, CO2 levels may drop drastically, as CO2 is absorbed into the (cold) ocean.

It is now well recognized that ocean temperatures rise during D-O (Dansgaard-Oeschger) events, followed by CO2 a few centuries later.  (I suspect that the CO2 increase comes from the warming ocean by desorption, but I have not checked this plausible belief by detailed calculation.)