United Kingdom

Look into micron filters or even submicron hollowtube filters. They're washable and reusable, they filter out most suspended solids, some heavy metals, and even pathogens (most bacteria and parasites, even some viruses).

I don't think they'd make grey water potable necessarily, but if you're worried about bathwater contaminating your garden then combining these with switching to biodegradable soaps and solvents is probably enough.

Adding a charcoal filter stage might bind some stuff that the hollowtube misses. You can even run it through a UV chamber if you want another line of defense, but that's a bit slower and might be redundant, especially if you're just watering your garden.

There are also ceramic filters out there that are supposed to remove pretty much anything the hollowtube filters would miss, but they're more expensive and the parts aren't quite as reusable. Supposed to make pretty much anything potable though.

I'd have to do more research about dissolved particles, but I think some of the filters mentioned can catch some of it. Otherwise it would require distillation or reverse osmosis, both of which are slower and require more expensive equipment. Building a solar still is an option but it's the slowest one.