I would say yes they will. However the crucial part is what will span the 4x4's? I would use 2x6's or greater for the spans. Then I would place the 4x4's inside the square instead of outside the square. They will then be more supportive being inside. That will eat up some floor space.
Then I would also on every 4x4 make 2x6 angle supports going from each post to angled up to the spaning lumber. This would make it even more stable.
The usual problem with these forts is they get soo high that the post area by the ground starts to rack or lean. I would also do this, about two feet from the ground I would nail 2x6's parrallel to the ground all around, and are you sinking the 4x4's in the ground? This will help also, but make the fort hard to move.
These things are fun and safe if you make all considerations when building.
I did one for a grandson last summer and it has become quite the attraction.