After the seedlings have grown to anything between the size of a thin spring onion to a pencil, you need to transplant them. Leeks are one of the most unusual plants to transplant and while there are hundreds of different theories about how to do it best (trimming/not trimming, holes/trench, sow-direct/transplant etc) most vegetable gardeners do it the way which is shown in the videos below. It provides a longer white part of the leek which is, let's face it, what the kitchen is most interested in.
All the gardening books will tell you to transplant them when they are as thick as a pencil, but if they have been sown very close together they will never get that thick, and you have to transplant when you think the time is right and they are big enough to handle.
This first video from Allotment Diary shows you the correct way to plant leeks. You have to dig large individual holes, drop the leeks in, and DON'T fill in the hole. The gardener in this video is using lovely thick specimens so it is easy to see the technique. If you want to find out how he grew them so thick and strong, see the video in the Growing Leeks From Seed resource here.
This video from Claire's Allotment shows the same process, but with smaller and slimmer seedlings, so if you're looking at your seedlings and thinking they're too small, you can see the same thing applies anyway.
