I find dill does better when it is sown directly out into the garden. I have some growing in the house right now, just because I saved alot of dill seed from my dill patch last year.
Might I suggest a fan in the room? It helps makes seddlings stronger.

Dills like cool weather. I have both direct sowed and started in cold frame.
There was not much difference. They beat parsley and chives which
are also cool loving.
If you dill seedlings are pale, perhaps they nee more ligh and/or fertilizer.
But the seedlings grow slowly. You have to be patient.
It is not necessary to to start dills inside. Just wait until the
temperature range is 38-to 50 and direct sow them in garden.

It is going to be warm enough this week that I am going to start hardening off my cool weather plants and moving them to the greenhouse. My dill is one of these plants.