Re: Cosmos from seed.
Pity about your lovely Cosmos, Dongle.
I was going to say about the light source, but then saw you already found that out.
In my experience, the trouble with window sills can be twofold.
1. Depends window sill you are talking about. Is a room you are using all the time, and the lights are on all evening after dark? Otherwise, they do not get enough light if the light source disappears suddenly and they are ind darkness for perhaps 12 hours or more, until the sun comes up again next morning.
2. Are they behind a net curtain, or up against the glass in full sun?
Turning the tray is good to stop them bending towards the light, but they bake when the sun is out, and cool rapidly as the sun moves round.
In other words, you get temperature fluctuations, the gentle warmth they need is not stable.
My guess with yours is, the temperature was too high, and that is what made them germinate so quickly.
I have a packet of Cosmos here at the moment, and it says:
They need a temperature of 15-20 degs, and to keep them moist.
Seedlings usually appear within 14-21 days.
When large enough to handle, transplant seedlings to 2" apart, and grow on on Cooler, but not cold, conditions.
You can plant them direct outside, but they are frost sensitive, so can't plant till end April - end May to be safe.
Personally, I think if it were me, and mine were not strong enough to support themselves, I would discard them and start again. You have still got 2 - 3 weeks to plant indoors, so don't give up.