It really sounds like the grass isn't growing well in that site? Is there a lot of shade?

Maybe you could find a more aggressive grass that would crowd out the weeds?

The part about removing moss caught my eye. Grass and moss grow best in completely opposite conditions: moss likes infertile, moist to wet, shady, acid soil. Grass likes fertile, normal to dry, sunny, neutral to alkaline soil.

So if moss was happily growing in your lawn area and the grass is so sparse that weeds are taking over, you likely are trying to grow lawn in an unsuitable site (shady, wet, infertile or acid). Identify what the problem is and correct it if possible -- remove trees, fertilize, improve drainage, add lime. Or choose a type of grass that will tolerate the condition. If you can't correct it, and there is no grass that works, then give up on lawn -- create a flowerbed or woodlawn garden or ...