Easy, the first piece of actual observed, measured evidence is that CO2 absorbs 15 micron IR. That single fact is sufficient to prove that CO2 is a warming influence. Quantifying the amount is more difficult to discern but it is certainly enough to prove the direction.
And yet more evidence that you are just another dupe...all that is evidence of is that CO2 absorbs 15 micron IR...look at the other side of that CO2 molecule and you will see that if it didn't lose it immediately due to a collision with another molecule, then it emitted it on to cooler pastures. CO2 has no warming influence. Only the most profound sort of idiot would believe that any substance that increases the emissivity of anything would result in warming...get up and go look in the nearest mirror and ask yourself seriously just how stupid you really are.
It seems to me that you are the one who is profoundly stupid. You don't grasp the implications of what you say.
Without CO2 all the 15 micron IR produced by the surface would fly away to space at the speed of light. You cannot expel energy faster than that.
Instead, all of the 15 micron surface energy is absorbed into the first few metres of the atmosphere.
This energy is then spread around by molecular collisions, transformed into various forms of kinetic and potential energies. The increase of kinetic energy is by definition an increase of temperature.
But this is a two way street. Energy in the atmosphere can also excite the CO2 molecule by collision in the same way as the 15 micron photon. CO2 molecules are constantly being excited by collision or absorption of a photon, constantly de-excited by collision or emission of a photon.
At lower levels this 15 micron energy cannot escape, there is not enough time between collisions, and the few actual emissions of 15 micron photons are reabsorbed.
At higher levels the air is less dense, the time between collisions is longer, the chance of a photon not encountering another CO2 molecule is greater, therefore more and more 15 micron photons start to escape.
If the amount of radiation escaping at the top was equal to the amount entering at the bottom of the atmosphere then there would be some truth to absorption and emission do not equal warming, although not totally. But emission is temperature dependant, unlike absorption. The temperature at the height where 15 micron radiation escapes is much colder than the surface, therefore less radiation is produced.
The difference between input and output powers warming in the atmosphere. Which in turn causes the surface to warm by reducing the temperature differential, which lowers the amount transferred by conduction.
A new equilibrium temperature at the surface is warmer because it has to radiate more in the bands of IR that still escape freely to make up for the 15 micron radiation that is (partially) blocked.