Are you familiar with CLIMATE SENSITIVITY estimates for CO2? I'll assume you may not be familiar with the technical terms in this discussion so let's start with "climate sensitivity" is basically the change in equilibrium temperature with changes in radiative forcing.
Here's the equation for this concept:
Basically it tracks the change in temperature (delta T) in the system as a function of the Radiative Forcing function for the item you are interested in, in this case CO2 multiplied by a climate sensitivity factor.
There are scientists who work in the field of GEOLOGY that have used paleoenvironmental data to let us know the general estimates of climate sensitivity for a variety of factors, including CO2. They have a reasonably good idea of the actual climate sensitivity for CO2 and right now most of the world's climate scientists feel that that number is high enough to account for a significant fraction of the warming.
If you would like to learn more about the topic here's a paper (may be a bit above your level but there are other sources).
The equilibrium sensitivity of the Earth's temperature to radiation changes