How to prevent loose brake cables with a single pivot side-pull mechanism?
It's because you have a combination of brake levers designed for V-brakes and side pull brakes. V-brakes need more cable pull for the same effect, so your levers pull the cable too fast. 

The solution is to get levers designed for caliper brakes, something like for example this (note "Caliper or Canti Brake Lever System"): http://www.tektro.com/_english/01_products/01_prodetail.php?pid=69&amp;sortname=Lever&amp;sort=1&amp;fid=3. There are also adapters for cable pull issues, but they are made to solve the problem other way round and cost as much as simple brake levers.