I would advise using a thatch rake, which can be purchased at almost any hardware store, on those thatchy areas. Next spring you may consider renting a dethatcher or a core aerator and aerating your entire lawn.

You could seed in perennial rye in those spots or convert those areas into planters where you can grow a groundcover, but keep in mind how invasive Bermuda is and requires garden borders. Grass maintained in the shade should be cut at a taller height and don't forget to fertilize with a nitrogen rich fertilizer. Water under those trees often as the roots from the trees suck a lot of water from the soil.

Where do you live?

Do you have water to the places where the grass is thin?

I live in Central Florida and yes, there is water to the places that are thin.

If you did not mind a mix of coarse bladed grass and fine bladed grass, then you could use St Augustine in the shade. St Augustine only comes as sod.

If you would like the grass blade sizes to match, you could use a shade tolerant zoysia like Shadow Turf. Shadow Turf comes in plugs only.

Well before you give up on Bermuda and live in Florida get a pallet of Celebration Bermuda to try in the shaded areas. It is shade tolerant, well shade tolerant for Bermuda and is widely available in Florida.

I like the Celebration Bermuda idea - thanks

Tifgrand is another shade tolerant Bermuda that you might want to look at. It's relatively new, but the growers claim 60-70% shade tolerance.

Another thought is to raise the canopy of the trees or thin them out so that the regular bermuda will not be thin.