I have never grown mint from seed. I always get a little start instead. But I was pleasantly surprised you planted them in small pots! That is usually the biggest mistake.
What I am seeing is too much water. Part of a healthy soil environment is air and drainage is critical. Not just having a hole at the bottom is enough however. The roots of the plant need to fit the amount of soil. Those tiny sprouts are doing nothing to suck up water. Stop watering so deeply for starters. Just concentrate on keeping the surface moist, moist not soggy. Those plants aren't using the water below 1/4 inch of the surface. I use a spray bottle to moisten the seed bed in pots that are 1" X 2".
Have you fertilized? Hopefully not, yet. Wait until they get their 3rd set of leaves and use a bit of OSMOCOTE 14-14-14. Only use a pinch and when you up pot don't use anymore...just transfer the pinch along with your plants and soil into the new soil. When they get 6" high give them another pinch. This last pinch should last 3 months. Mint is voracious and grows quickly. We can help you decide whether or not and when your plants need more. Just be very sparing. The seed has energy enough to get them started.
I need to know what soil you've used. Hopefully it was potting soil but sometimes potting soils have added stuff that really shouldn't have been added; fertilizer and those water holding gels/sponges. Could you check? Take a picture of the bag, both sides to send?
Stop watering every day and just use a spray bottle to keep the surface moist. You might have to transplant with fresh soil to get rid of the soggy soil. No fertilizer, need to know if they put fertilizer in that soil. If there are water holding gels/sponges I'll talk you into getting some plain old potting soil without!
Light? I'd use a south facing window, not east. Because a plant is genetically voracious and grows fast it needs some serious photosynthesizing and the first ingredient is light, the next is moisture and then properly balanced fertilizer but always on the spare side. Do you have a covered porch? I'd take these guys outside on the covered porch and let the wind and higher temperatures dry up your soil. Your soil might just be too wet but for now let's see how they do. It takes time but this is the slowest LOOKING part of a mint's life so enjoy it while you can! Keep your babies out of direct sunlight but they need lots of light. Are these to become indoor plants? What are your thoughts? Where is it you live? Do you have a garden, a yard? Mint can most certainly get out of control but if one is out poking and pruning and fussing in their garden often, this just won't happen.
If you want your mint to be planted out in your garden you'll need to acclimate it to the sun FIRST. This is done by gradually increasing the time in the sunlight. We can lead you through that. But if these are to be indoor plants let's stay in a south facing window and/or a covered porch. Summer time I always take my indoor plants out on a covered porch to get better light so they can make more food for themselves, get hosed down of dust, and when they come back inside for the winter they look brand new. Please keep us updated as you grow your mint...so important to have successes so that you'll continue to 'grow' as a gardener!