Grab yourself a solar oven....it is a parabolic reflector with a focal point at which one can cook....put a thermometer at the focal point, and a thermometer a few feet away from the reflector...point it at clear sky away from the sun and watch the temperature drop below the ambient temperature....take it out on a clear evening when the ambient temperature will not be above 45 degrees and won't drop below 40...point it at clear sky and hang a bag of water at the focal point....come back in a few hours and note the ice that has formed while the ambient temperature was well above the freezing point of water....
If back radiation exists, why then, does the temperature at the focal point of the reflector drop below the ambient temperature when it is pointed at clear sky that is, according to you and yours, chock full of CO2 back radiating energy and warming the earth
To expands on Ian's correct explanation, whatever object you put in the center is sort of like a point source at the focal point. The energy it radiates in all directions strikes the parabolic mirror, and is all bounced out of the oven. The backradiation, being diffuse, is not focused by the parabolic mirror. So, energy goes out from the focal point, little energy comes in, temperature drops.
Actual observation there...
And the observation is 100% consistent with how backradiation works.