Now that I see the photo, I think you should trim the plants, before they produce seeds. It takes time (as long period), but it is simpler than taking every plant apart.
I would in any case remove the larger weeds manually, as soon as possible.
Weed killer could also help, e.g. with a total reset of garden, but I would use as last resort, but maybe you can do it for one part of tour garden this year, an other part next year, etc. But remember to remove seeding weeds regularly. This should reduce the pressure of weeds.
You should also research your garden. What weed in what part, so that you can better design your garden (plants who requires wet, sun, shadow, sand, rich soil, ..). Weeds are good indicators.