How can I discourage ragweed from growing?

Have you tried covering the area with clear plastic when your wrea is getting some sun or when it is hot. I would try 2 weeks for a starter.

Do not let them go to seed!

(Info from Purdue University)

Common ragweed plants allowed to grow for the entire growing season can produce 32,000 to 62,000 seeds per plant.

But if you don't have time for all that hoeing, consider that you may want to be lazy later in the season:

Plants that emerge mid-April through May can produce up to three times the seeds and biomass as plants that germinate in mid- to late June.

And it might actually be helpful to stop tilling. Ragweed seeds can survive a long time in the soil (one source said 80 years (sorry, I forget where I read this), others say 20-30, even up to 40 years). According to the Purdue sheet above, the deeper seeds won't germinate without tillage bringing them to the surface. Unfortunately that reference doesn't help you with control strategies in your home garden much.

They are hard to kill, too. You can break off the stem and it will regenerate. This means mowing won't kill it, unless you're mowing frequently enough to starve the roots. If you're hoeing, make sure you chop it at or below soil level; even here it seems like there's a chance that an established plant can regrow. Pulling them up by the roots seems effective, especially if you keep up on it and get them while they're small. And if your garden soil is fairly loose, it's not too hard to yank them out.

Seeds can be dispersed by wind spreading them along the soil surface. This means that if you have uncontrolled ragweed in another part of your yard, have neighbors with uncontrolled ragweed, nearby fields, or if your town doesn't keep the roadside ditches mowed, then you'll end up with more seed in your garden year after year.

The strategy I would recommend is to reduce the ragweed seed bank:

• Don't let any go to seed. This is critical. Every plant that goes to seed makes your job next year that much harder.
• Be vigilant. Go out every week and pull out all the ragweed seedlings in your garden you can find.
• Cultivate (shallow hoeing) weekly. This will chop off small seedlings before they can establish a good root system.
• Keep down any ragweed plants that are within your zone of control -- keep your yard mowed (which won't necessarily keep the ragweed from growing, but you can at least reduce the amount of seed) and pull plants where you can.
• Don't till deeply if you don't have to. This will bring more seeds from deep in the ground to the surface where they can germinate.
• Be aware of where your soil amendments are coming from. E.g. if you bring in some local horse manure where hay was used for bedding, or if spoiled hay is mixed in with the manure, you may be importing ragweed (and tons of other weed) seeds. If you buy compost, beware that it could be contaminated with seeds. Above all, don't use hay for mulch!

Unfortunately this strategy can take a few years to really pay off, and if you let it get out of control for one year, you have to go back to stage 0. On the bright side, this is a good general weed-control strategy, so you will reduce the pressure from other weeds at the same time.

If it's a shaded part of the garden, you could grow something that likes part shade and grows pretty big, like turnips, kohlrabi, kale, brussels sprouts, cauliflower or broccoli and mulch the pants off of that section.