First – don’t use soluble fertilizer of any kind – the weeds can use this better than the grass can. Then, use the aggressiveness of the grass to your advantage – water deep and mow high and often – even when it doesn’t seem like you need it. High means 3-4”. Weeds will outgrow the grass tall-wise, so when you mow, you’re giving the grass a little trim while cutting the heads off the weeds. (read that analogy somewhere, can’t remember though).
The weeds’ll have to expend more energy to heal and regrow while the grass keeps trucking. After time, the grass will gain the advantage as weeds are smothered by little light and depleted energy for regrowth. Now, some weeds can take this – they’re probably other grasses. Sandburs for instance will take a laying down attitude to under-ride the mower. Those weeds are best just pulled – and try to do it before they go to seed. I’ve also read that corn meal can help too. Mycorrhizae can be a big help – grass can form a relationship with this fungus while most weeds can’t.
The fungus will then mop up nutrients that the weeds need and boost the grass’s competitive power even more. Also, try to find a grass that grows well in your area. IE – Buffalo Grass grows great there in Central Texas. Bermuda does so-so. St. Augustine really doesn’t fair so well. Etc… Key to success is patience. It can take a few seasons, and you’ll still have to do some pulling. We’re doing the same thing here. It takes time – but the grass is really agressive when it gets a foothold and quickly takes over. That’s why it itself can be called a weed.
Have you ever thought of contacting some 4 H kids with sheep or goats and allowing them to graze for a few days and basically cut the ground down so close that you could then use a rotatiller then re-seed? Or then use a lawn mower?