What is the weed? We need photos :-) Some weeds - like horsetail, bindweed or knotweed - can be nearly impossible to be rid of without virtual constant attention.

The last picture appears to show Sheep's Sorrel, Rumex acetosella mixed in with seed capsules from some member of the Brassicaceae. But that is not what's in the first two pictures.

Agree that it needs to be dug out. But I'd use a fork, not a trowel. A hand fork if you are not very flexible, a digging fork if you're ok bending. You can get a better purchase with the latter and it saves your hands and wrists.

I'm not familiar with this weed, but is smothering it with cardboard and mulch a possibility as an alternative?

That could well work but you'd not be able to plant vegetables in the bed immediately if it was covered in cardboard,

How big are these raised beds? Regular weeding will keep even plants which run on stolons in check. You can worry stuff to death by pulling it repeatedly. Just keep pulling them and eventually you'll have them under control.

Agree that you are fighting an uphill battle, as I too have rumex and have only recently begun to discover it everywhere. Moved into the house summer of 2018 and it definitely took advantage of my ignorance that first year. Persistence is the key. At the last house I had creeping charlie, another spreading weed. I had a huge old cement double laundry sink that we sunk into the grass, filled, and used as a planter with only the original tiny size drain holes at the bottom. It took 3 years for the creeping charlie to get into the sink, but once it was there it took over. Roots/rhizomes crept over at least 3 feet across to get through the tiny drain hole and up another 2 feet to get to sunlight. Good luck.

Agree that the plant in your first two pics is not the same as the plant in your third pic. I am not an expert, but look up pigweed/amaranth species and see if you think that fits. If that is the case, it spreads by copious seed production and you could get ahead of it by just making sure it never produces flowers.

Thank you all so much for this excellent detail on the weeds that are growing in my garden beds and yard - and to /ken-adrian for the instructions on combatting it (even though it will be a very slow process).

To answer one question - my raised beds are 4’x8’ and 12” deep (I had dug down about 2 - 4” lower than the bottom of the garden bed so I could add enough soil to be able to plant vegetables with deeper roots.

i think I’m going to give the raised beds a break this summer, and use a few elevated planter boxes to grow our favorites, and then work at removing the roots from my garden area.

i had hoped to avoid using weed killer, but it looks like I have no choice to use it on the weeds just outside the garden because they’ve creeped back again after just a month!