The exhaust of a rocket is a force exerted by the rocket itself.
Therefore it will require a body extrinsic to the rocket for it to create a force pairing and thus produce motion.
Maybe according to Papa's imaginary laws of physics, not according to Newton.
According to Newton (and reality), there exists a force pairing between the rocket and the exhaust.
The rocket pushes the exhaust backwards while the exhaust pushes the rocket forwards.
No external entity required.
If you like, you can consider the exhaust the "extrinsic body".
So rocket exhausts do not exert a force then?
No, they do exert a force, on the rocket.
What happens later is fairly irrelevant for a rocket.
Just like when the bullet hits it's target and is slowed down, the gun doesn't magically recoil as if there is some magic connection between the bullet and gun; the same is true for the rocket. The rocket's exhaust pushing against something wont magically make the rocket go more, at least not once it has left the rocket.
Yeah, you don't understand anything, let alone Newton's third law.
There you go projecting again.
Here are a few nice simple questions for you:
You are out in space, you have the rocket and the "exhaust" trapped together.
The exhaust then starts moving out the back of the rocket.
What force causes the exhaust to accelerate?
What is the equal and opposite force required by Newton's third law?
What is accelerated by this latter force?
Or even simpler, just one question:
What causes the curve in the photo in the OP?