I can't jump over the 2 tile gap :(
I've tried spamming jump, holding direction and jump, pressing the jump at different times.
I got over it once but couldn't repeat it.
It looks like we need to further improve that, since you are not the first person to have the issue :/
What you have to do is get in "running" mode, that is walking twice in the same direction, the player sprite will change when this happen. Jumps will then become long jumps.
I think the "turn-based" / puzzle nature of the game may not be readily apparent, probably because enemies don't move.