The Brookfield Zoo can get pretty crowded, but due to its size it generally handles crowds better than the Lincoln Park Zoo (really, better than most Chicago Area museums). Crowds are worst during the first few beautiful weekends of the year, although they can be pretty bad on any good weather day during the summer, particularly on weekends.
The main problem spots are getting in (parking as well as tickets) and getting lunch. Both are best solved by arriving early: getting to the zoo just as it opens gives you much shorter waits to get to the parking lot, and I always recommend getting to lunch early—certainly by 11:30 and on particularly busy days, I'll try to make it to lunch just after 11:00. Of course, you can bring your own lunch and skip the line entirely.
In general, the North parking lot handles busy days much better than the South lot which has fewer lanes to get in and can fill up. Keep an eye on Brookfield's Facebook page or Twitter page which will post updates and information when traffic or lines are particularly bad (the same crowd information will be posted to both sites, so you just need to check one). Even when arriving early, it can be helpful to bring a snack or activity to do in the car while you wait, waiting in a slow moving line of cars can frustrate even the most patient kid (and adult).
The North parking lot also has more windows to buy tickets, and generally has shorter lines. You can skip the line entirely by ordering tickets online and printing them ahead of time. If you're planning to buy a membership, you can order that online as well: your receipt will come with a link to print a temporary membership card.
For lunch, the easiest way to avoid crowds is to bring your own and eat it outside. Of the permanent restaurants, I think BZ's Red Hots is the best for crowds, but if you're in early, Café del Sol is fine. For the summer-only restaurants Bison Prarie Grill does fairly well with crowds as well, just know that you'll need to eat outside. See the food page for more information on those options.
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