"Seed-free" straw doesn't exist.

It's easiest to deal with the seedlings when they're small -- rake them out, Hand pull/weed. Or cover with 2 inches of mulch.

I suspect you are using hay, instead of straw. Hay is supposed to have seed in it, straw isn't. Straw is the stalk of cereal after the grain has been removed. Hay is long, dried grasses with seeds intact. I bed my chickens down with hay, because they like to pick through the seeds. I put mulch my garden down with straw. I suspect you don't want to hand weed a whole flipping vegetable garden, and from the looks of your photo those are vegetables. Depending on the size of your garden I suppose mulch is an option, but if it's large not a very cost effective one. There are pre-emergent weed killers for that use, but I grow my veggies organic and assume you might too. Yes, you can do the plastic tarp routine, but I dislike it as it also kills beneficial soil organisms, and earthworms don't exist well under it either. I'd suggest if you don't want to use a cultivator, then indeed find a different source for better quality straw.

Hay is an animal fodder and most of the time it will be greenish more than tannish, unless it is really old or spoiled. The only reason hay might have seeds is because it was cut late, after the plants developed seeds, which is not normal. Hay cut after seeds develop is of poorer quality for fodder.
Straw is the residue left after harvesting various grains such as Wheat, Oats, Barley, Rye, etc. and some of the grains can be left on the stalks and these will not be "weed: seeds but seeds of whatever the grain was. There is straw now being sold here the at is seed free but that is about twice as expensive as a regular bale of straw and is wrapped in plastic. I would not be concerned about any seeds that are on the straw stalks.
Covering the existing "weed" seeds and denying them access to the sunlight they need to grow will kill them. Solarizing, even in plant hardiness zone 10, is best done between June and September.

and this is why i only used straw once in my gardening life ...

you simply need a better mulch..

most of the weed seeds need to be on the soil surface.. and in sunlight to germinate ..

a proper mulch.. 2 or 3 inches deep.. will bury them.. and the vast majority will not sprout ...

you are using the wrong product for the wrong application ...

ken

It is not at all abnormal to find some seed in hay, especially in mixed grass hay. They don't all follow a schedule on exactly the same days and week to develop seed. It's preferable if the seeds are immature, but the reality of it is that you harvest hay more with the weather in mind and it's a lot more preferable to have it vary a little bit in maturity than try to produce it against the weather. The better test for quality of a bale is to inspect the leaves and stems but it's a moot point if you're just going to dump in on a garden. And you're gonna get some seed regardless of quality. Like Ken says, pile it deep. Those grasses who do emerge are etoliated and easy to pull.

Hay is a broad leaf plant. Straw is in the grass family. OP has straw seedlings.

Jean, hay is grass as well as straw.

Here is a link that might be useful: hay

also ... this is one of those mulches.. that if you do want to use it... you have to pile it ridiculously high ... so that as it settles.. you end up with the depth you want ... or need ...

just spreading an inch or two ... and you will end up with the bare soil the seeds need ... but to get a good cover.. you would probably have to put down 4 to 6 inches .. and it will look silly until it settles ..

and by the time you put down 4 to 6 inches... your investment.. is going to be equivalent to a better product ....

we all struggled as we went thru various mulches.. until we found the right one ... i DID try straw ... on strawberry .. wonder where that came from.. lol .. and i had the same weed issue ... i was growing a grass patch .... so i gave up on strawberry ... that and it wasnt worth the 11 months of care.. for the little result i got ...

ken

Just want to thank you all for your responses back to my ongoing issue of growing some very healthy weeds these days! Ultimately, I think I just need to get another source for my "straw"..
I've been doing this gardenThing for about 10 years now, and in most of the years previous this has not been a problem. Here is the usual scenario.. I wait until my plants get large enough to wind a soakerHose up and down the rows without breaking them off.........then, lay down two sheets of newspaper on top of the hose...... following with the "straw" on top of that.. There are always some weeds that poke up through the newspaper and straw and I expect that, but the past year or two has been extreme. I've had weeds growing much taller than my pepper plants!
So, like I said..........ultimately I need a new source for the straw I've been using. I'm open to any ideas of something that might work better if you guys have any suggestions on this.. Thanks again in advance for any responses back....

Around here, 'hay" refers to "alfalfa hay."

Try pine shavings or leaves/chopped leaves.