step 1 - stop tilling, tilling brings up dormant weed seeds. some weeds can stay dormant for a long time. yellow star thistle seeds are known to germinate after 20+ years in the soil.
Perhaps, but consider this. When plants produce seed, it usually either falls to the ground or it flies away in the wind. Sometimes it is carried by critters or birds, but in any case it is destined to fall to the surface of the earth, where it will germinatre when the conditions are right.
There is many times more seed on the surface than down deeper where a tiller may leave it. If left deep, it will not germinate. Yes, some seed will last a long time buried, but many types will not make it through one season if buried.
I like to till my garden. It chops up all garden remains and weeds and incorporates the organic matter into the soil where the microbes quickly reduce it to plant nutrients. It introduces oxygen into the soil for the microbes and speeds the process of decomposition. It loosens the soil so the roots of new seedlings can easily penetrate the soil.
If you want to use a tiller to help with the weed problem, till, then wait until the area turns green with sprouted weed seed, then till again, but shallow. This kills the sprouted weeds without bringing up many dormant seeds from down below. You can now plant your garden without getting so many weeds.
Yes, you will get some weeds. The thing to do is take the hoe to them while they are still very small. It is a simple matter to hoe a small garden while the plants and weeds are still small, and especially if the soil has been loosened with a tiller.
I have a pretty large garden. It is not practical to mulch it all, so I plant rows wide enough that I can take the tiller between the rows when the plants are small and get those small weeds. This tilling is shallow, maybe an inch is all. I then hoe between plants in the row. Some places I have to hand pull the weeds.
One thing you need to know about mulches, a mulch is food and hiding places for snails and slugs. Mulches are also sometimes loaded
with weed seeds. Depending on where you get the mulching materials, you may also introduce new varieties of weeds into your garden with the mulch. This is a very real possibility if you get hay or straw from a farmer.
It is not my intent to start a till or no till war, but want you to just consider these ideas, then try something. You will after a time develop a gardening method that works well for you.