Compare the ionization energies of C and Cl?
It's the other way around,


moving left to right across a row increases the IP due to increasing nuclear charge, and 
moving down a column decreases the IP, due to increasing shielding of the nuclear charge by core electrons.


Chlorine is only one row below carbon, but it is three columns to the right.  As the following chart illustrates, moving to the right by one position generally has a similar or larger effect than moving down one position.


(image source)

Therefore, in this case moving further to the right by 3 positions (and increasing the IP) strongly outweighs moving down by only one position (and decreasing the IP).  Hence, in this case the IP of chlorine would be predicted to be greater than the IP of carbon.  Indeed, the IP of chlorine is reported to be 13.0 eV, while that of carbon is only 11.3 eV.