How to remove weeds from garden?

Probably because you chop them. You're creating more weed plants. You need to get rid of the grass, and then use a barrier between your lawn and garden to stop the grass from invading your garden. Then use a lot of mulch to suppress the weeds. If they come up through the mulch, spray the green part of the weed with full strength vinegar. Do not get it on your good plants. Use a paper funnel to direct the spray only to the weed. Eventually they will die.

try smothering them out with newspapers,cardboard,heavy landscape fabric and top with mulch

Put bown brush killer, but you will not be able to plant there for a few years, read the bottle.

Install a root barrier all around the garden.

What I’m wondering is what kind of weeds are these, I don’t think I‘ve ever seen a weed with 5in thick roots. If the other suggestions don’t work, I think I would find out what type of plant this is, contact you local nursery or agricultural center to get help.

Is it in rock or grass? We had a rock front yard and I got tired of pulling the weeds and / or trying to find an effective but not hugely expensive (think landscaper) way to get rid of them. We bought a weed burner. Ran it off our grill propane tank. only took out the weeds we wanted...simple point and shoot a small flame at them. One of us was standing by with a hose but never had a problem. Didn't have to worry about chemicals with my small dogs. after the first time, we had to do it again after about 2-3 weeks, but already there were fewer coming back ...did it again about a month later, again fewer coming back... Worked great for us.