If I try to push the crate away from shore, it'll gravitate again towards the shore for some reason and I'm back at square one. I can try to jump on the crate, but I'll end up slipping down into the insta-killing water.
What should I do here?
Drag the crate backward a few feet until it's underneath a hanging rope. Climb the rope and make your way up and through the trees, until you reach the far upper right corner of the forest. Here you'll be able to push over the top of a dying tree, giving you a longer bridge to push into the water. Use that along with the crate to cross the gap.
Here's a video of the process:
