I bought my house in winter and found in spring that my yard was a mess....
You don't give any specifics so I can't give specific advice, but will share what I learned (the hard way).
You need to focus on 'growing' the roots of the grass first. Mow low an often. Water deep when needed.
THEN you can kill the weeds and, of course, then seed in the bare spots.
Two times a year isn't right. I 'treat' my lawn 4 times a year; weed killer, fertilize x2 and winterizer. And I'm not over concerned about my lawn, so I only do what I have to.
Oh, and I use grub/bug killer every 3 years.
Don't fertilize in spring before the ground is 60*, or you're just fertilizing the weeds.
If the above doesn't show improvement, do a soil sample test.