The problem could be a combination of factors, but in my experience most lawns struggle in summer because of the wrong grass species choice. If you live in a very hot and dry area, common grasses like tall fescue or ryegrass don't handle heat well. You should consider whether your lawn was seeded with varieties meant for temperate climates instead of extreme summer conditions. Then there's the water issue: watering frequently but lightly doesn't help at all, actually it makes things worse because the water doesn't penetrate deeply and roots stay shallow. Better to water less often but more generously, maybe two or three times a week depending on how hot it gets where you are, so the water reaches where the roots need it.
As for the soil, it's important to understand whether it retains moisture or if it's too sandy and water just runs through. If you notice water running off easily, your soil needs organic amendments like compost or peat to improve water retention. Then there's the sun factor: if your lawn gets direct sun from 11 to 5 in the afternoon, during summer it's almost impossible to keep it green without wasting water. In that case you could think about lowering your expectations or converting those areas into flower beds with tougher plants, which by the way are less demanding to maintain.
Honestly, if I were in your situation I'd get in touch with a local nursery that knows the climate of your area well and have them recommend the right seeds. Maybe it's time to start fresh with the right grass variety, because continuing to fight against your climate's nature is a losing battle and a waste of time and money.