You've got the answer. Not enough light and they are stretching their little stems to get to it. Any way you can rig up those long fluorescent light bulbs about 4" above the tops of the plants and leave on about 18 hours a day? You could use a double fixture and put a warm white bulb in one and a cool white in the other. Turn the plants once a day. I have two shelves on the wall in my laundry room. The light fixture hangs from hooks on the bottom of the top shelf, with a chain at each end that I can shorten as the plants grow. You can get the light fixtures that plug into the wall as well as the ones that get wired into the ceiling. The plants sit in the shelf just below the lights. Take a look at these sites.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.