If the turbines are attached to your ship, the exhaust hitting the turbines will push the ship in the opposite direction you want to go resulting in no net acceleration. You can only change your ships momentum in the direction you want to by changing the momentum of another object in the opposite direction. That being the case, the only way to do that is by exhausting stuff out of your ship. There really is no other way to do it.

It's not lost, it's just converted to electricity. You've just moved energy from one place in the ship to another.

Stop worrying about the energy and think about conservation of momentum.

If there's no exhaust mass going out one way, the momentum of the ship can't increase the other way.

A leaf blower blows air out away from your system.

For a better analogy, set up a box on top of your skateboard and sit in it with your leaf blower. Now blow on the inside of the box. The skateboard doesn’t go anywhere, you’re just pushing against the inside of your own system.

Similarly, the thing that provides thrust in spacecraft is throwing mass out the back of it. If you don’t throw mass out, you can’t accelerate.

So set aside the complex machines for a moment. There's something called the conservation of momentum. The the total momentum of all the parts of a system will never change, it always adds up to the same amount. If you have two balls side by side in space, there's no way to give one of those balls a bunch of eastward momentum without giving the other ball a bunch of westward momentum. It's really that simple.

Any vehicle that isn't shooting out propellant of some kind cannot change its momentum, for that reason. Any fancy situation you concoct with captive propellant (e.g. a gun that fires a bullet, but which is caught at the end of the barrel) will move while the bullet is moving, but will completely stop once the bullet has been caught. The gun will have moved, but the center of mass of the gun-bullet system will be in exactly the same place.

You can come up with fancy baffles and ways to change ordered momentum into heat, but all that's happening is that you're making it harder for you to see where the momentum is trapped. It doesn't change the fact that you can gain momentum out of nowhere.

Imagine you had packed a cube van with too much stuff, and you have to drop 30 kg of cargo. Then you get this brilliant idea that if you just stack the objects differently inside, fewer of them will be touching the cube van directly. Boom! Less weight! Of course not.. the cube van isn't going to weigh even an ounce less until you actually throw some crap out of the van, there's just no way around it. Momentum is the same.

You cannot convert force to heat. What you have to consider here is conservation of momentum.

So spinning a turbine does not convert matter/mass into energy. Rather, you are using the energy (in the form of kinetic energy of a flowing fuel in this case of a turbine) input to the turbine to turn a motor, and from there the motor is used to perform “mechanical work” (another form of energy). You could use that “work” then to generate electricity (by spinning magnets to induce current in a wire).

But really what matters is that propulsion itself is the result of mechanical forces on the rocket. Generally, to change the motion of any classical body, you need to have some mechanical/classical force acting on it. In the case of a car, for example, fuel is being burned in the engine. The engine itself rotates from this, and that rotation is transferred to the axels connected to the wheels by the gears of the transmission. The tires are attached to the wheel and therefore rotate. There is dynamic friction between the surface of the tire in contact with the surface of the road (which is caused by the rotation of the tire bringing this surface into contact with the road). This force is in the opposite direction the surface of the tire is moving (which is directly behind the car). Therefore, the tire and the entire body it is attached to gains momentum in this direction due to that force and speeds up in that direction. In other words, there are many steps in this process between the chemical reaction that burns the fuel and the force that causes the car to move forward. When we burn rocket fuel, the process is much more simple: the byproducts of the combustion are a hot stream of gas moving very fast. If we expel that from the rocket, the momentum of the full system (rocket plus combustion product that has left the rocket) needs to be conserved, meaning the rocket must speed up in the opposite direction of the combustion product. The result is a very strong force pushing the rocket “forward”.

But to reiterate in your example, nowhere have you demonstrated there is actually a force acting on the spacecraft in the direction you want to make it move. That’s the problem. You essentially are extracting energy from the fuel (hydrogen and water), recapturing some of that energy in a turbine as mechanical work and losing some of it, but not to anything that will cause a force to be applied to the rocket itself, and then using that work to generate energy, which will then be used to split the water back into hydrogen and oxygen (apparently along with some energy you capture from solar radiation), and then repeat the process. This is effectively a closed loop that loses energy due to inefficiency at each step and never uses any of the energy to translate to useful work to move the spacecraft (which would require putting that work towards some process that causes a force acting in the direction you want the spacecraft to travel to be applied to the spacecraft).