When I first got my mare, she would panic and half rear if she stepped on her lead rope while grazing. She never pulled hard enough to break her halter, but definitely freaked herself out. These days she’s totally chill about stepping on it – she keeps her head low and backs up carefully to figure out which hoof she’s standing on it with so that she can release the pressure and keep grazing.
Here’s what I did: I first taught her to drop her head on cue in response to me pulling down on the lead rope directly under her chin. Once she was comfortable with that, I let her graze in an enclosed barnyard (so she couldn’t go anywhere if she panicked) with me holding the very end of the lead while the rest was on the ground. She stepped on it a few times, had less of a reaction every time, and then it was a non-issue. It helps that she’s a glutton and would much rather graze than panic!