How to fix my backyard?

Maybe you could do some ground cover and place pavers or stepping stones.

Are you giving the grass seed something to grow in. From the picture it looks fairly muddy. Sprinkle the grass seed and cover with good soil. Home Depot and Lowe's carry bags that are not expensive of all purpose soil. Keep off of the area until the grass spreads. Also read the package of seeds. There are special varieties for shady areas.

https://www.gardeningknowhow.com/ornamental/trees/tgen/tips-for-growing-grass-under-a-tree.htm

https://www.pinterest.com/explore/no-grass-backyard/

https://blogs.reading.ac.uk/grass-free-lawns/rethinking-the-traditional-grass-lawn/

If you are set in wanting to have grass in that area, I suggest contacting your County Extension agent for advice about what types of grasses grow well in your situation (shade). There are certain species that do better in shade. Otherwise, I suggesst some other type of ground cover that does well in the shaded area and the Agent can provide you ideas for that as well. Their service is free for the asking.

I just finished mowing almost an acre with a push mower on a hot humid day, so I'm not too big on grass right now. I would plant hostas on the sides and make a pathway with moss & pavers down the middle.

In my opinion either the large trees or the grass has to go. As long as the trees are there, it will a loosing battle to try to grow a lush carpet of green lawn.

If you really have your heart set on it, then section off areas, and re-plant

with new sod. If not, then go with other non-lawn landscaping ideas materials, such as paving stones, bricks, pea gravel, and rock.