For an exponential increase to show up temperature would need to linearly relate to CO2. However, temperature basically never linearly relates to anything. For the simplest example suppose you use light to heat a sphere in space. Double the light and you don't double the temperature because black body radiation is T^4. Doubling temperature takes 16 times the light.

As to earth, simple models are laughably inaccurate because the equator and the poles have wildly different temperatures and heat is moved between them though the movement of air and ocean currents.