I am currently looking after 3 trays of seedlings for a friend.
I'd had them 2 weeks, and hardly anything had grown. To compare, stuff I sowed after she had left them with me, is now larger than the stuff she had.
So on friday I scraped back an inch of the compost, and added a top dressing of my own sieved compost.
Overnight, they all started to look healthier and the melons all grew one extra leaf when they had done nothing for 2 weeks.
I suspect the issue is that the bought compost is full of large bits of woodchip, and this is taking all the nitrogen from the soil leaving little left for the plants. Or you are sowing too deep.
Or that this same compost is too big to allow seeds to get the right conditions for germination.
I've been sowing all my seeds on an inch of coir, over a couple of inches of my own compost. And potted into my own compost.
Try sieving the compost you are sowing seeds into, and maybe on a half inch or less sized sieve, even one of those small plastic ones you get in kits from Wilkos or garden centres. Seed sowing compost needs to be light.
If you buy compost that still looks like the thing it was, it isn't ready. Take it back and get a refund. this is why many bags have little corners torn off to see what is in the bags.