You have other options. You can use landscaping fabric as well, although it is probably more expensive than the black plastic.

I assume the size of your garden is outstripping your ability to read newspapers and to grow and mow your lawn and not because it quits working as the size gets larger. There would be other ways to increase your supply of these materials, though. Ask your neighbors for their newspapers and grass clippings, ask a kid who mows lawns in the area, etc.

You could also get used carboard boxes from a store and lay the cardboard down. It wouldn't need a layer of grass on top, although it would look better if it was there.

old carpet would also make a good ground cover.

The black plastic doesn't let water through very well, but the other options would let water pass through in greater or lesser amounts.

Plastic over soil can keep the soil from breathing and the soil can then become anaerobic and that will cause major problems with the plants growing there that have roots that need air. Plastic is not a substitute for newspaper, or construction paper, or cardboard.
Landscape fabric does allow some air into the soil, but it also creates other problems because often things such as Quack Grass roots will grab hold of that fabric and use it to spread around.
The simplest thing to do is cut the size of that garden by making several smaller beds, which are easier to care for also. There is simply a lot of wasted space in a 30 x 20 garden plot.

Hate to be disagreeable but I disagree that the simple size of a garden plot results in "wasted space." The garden I enjoyed the most of all I've had was about 40 x 25 and it had paths going through it. I didn't consider one inch of it wasted and thoroughly enjoyed walking through (not next to) my garden.

But to address the original question, and leaving preen out of the discussion... Anything covering the ground will "prevent" weeds. The results depend on so many things, it's hard to predict what results you in particular would have with them. For instance, if you have a seed-dropping tree nearby, you will have tree seedlings no matter what. However it's easier to pull an oak or maple seedling out of a layer of mulch than the harder earth underneath. Barriers don't stop seeds from above, usually. So the decision comes down to what you have available and your requirements for appearance, and whether you want to spend a lot of money or use "free" materials. Leaves, mulch, grass, newspaper, pine needles, it's all good. My personal preference is leaves or mulch over newspaper. I wouldn't consider your plot too big for newspaper. If it takes you a while to cover it with something else, bricks or rocks can hold it in place.

been there and done that, and outgrew the newspapers as well. if you live anywhere near the country, drive out to a farm and ask them for their paperfeed sacks! generally there are multitudes to be had and the farmer doesn't have to haul them to town to the recycling center! the sacks, when cut lengthwise to open, are twice the size of an open newspaper and are generally 3 ply. depending on the area i need to cover, i may not bother separating the plys, and they stay put better than newspaper ... or you can put a lite cover of mulch on them to help. water passes thru just fine and the weeds don't stand a chance. it's free and it's recycling and it's neighborly :)