They are being kept too wet, causing 'damping off', which kills the plant. You need to water from the bottom, ie set the pot in a saucer, fill the saucer with water, let sit for 15 minutes and then dump off the unused water. Allow to dry out a day or two before watering again - sometimes, it's a week between waterings.

Your soil should have some grit mixed into it, in the form of sand or perilite - this will allow the new roots to dry out and allow oxygen to reach them. You have to check them every day to see what they need - they're like babies - need constant attention. I believe it's worth every penny to buy already-established plants.

It's not wet that kills the seedlings in "damping off" syndrome - all seedlings need moisture - but the microfungi that thrive in damp, stagnant conditions. Sterilize the compost before planting. A neat way to do this is bake it in the microwave for 3-4 minutes if you only need a small quantity. Secondly ensure that there is adequate ventilation. If necessary put a fan next to the tray, though you can also get good results sowing out of doors during the summer. Better luck next time!

Aside from the good advice listed here, you may want to also consider electrifying the soil for anything that is having a difficult time surviving. It can help your plant get past any rough spots that you're having.

Soil electrification helps plants repair themselves from various forms of disease or stress, boosts nutrient uptake, and helps activate soil bacteria. Furthermore, particularly for your situation, it helps with the germination rate of seeds.