A lot of groups coordinate on discord; they'll have a template & an area they want to target. Some of the larger groups may have thousands of people. It depends on the group, but in one I've seen, they organize people by birthdate for some of the designs, and by color for the larger part of the design, and let people go.
There's a way you can get an overlay of the intended design on the r/place canvas, supposedly, but it didn't work for me. But like they'll have templates or whatever.
One of the largest groups is "the void" and normally it just tries to black out the canvas. This year it looks like they're helping out the fuckspez black and white design mostly.
There's probably an internal concept of turns, but it's not directly exposed to the user. You place a pixel, you wait 15 minutes, you can place another.
As far as larger works of art, it's a lot of people coordinating together and carefully crafting something.
It's one pixel per user per timeframe, not one pixel per timeframe - if that makes sense.
Outside of that, it's likely timestamped to the milliseconds on the server side, so it'd be pretty rare for two users to hit the exact same pixel at the same time and millisecond (or fractional millisecond). And even if, the server would still process one first, not both at the same time.
For some larger coordinations, there are bots that one can use to achieve swaths of changes very close together.