Generally engines turn the chemical energy in fuel to kinetic energy, in a helicopter that kinetic is turned into gravitational potential energy when it goes up. Now let's imagine a helicopter hovering above the ground at a fixed height. To stay where it is the engine has to keep supplying a force to counter gravity so there is kinetic energy being supplied to the system of helicopter and earth but the gravitational potential energy is not changed. Energy is conserved so what is that kinetic energy being turned into?
[Physics] Conservation of energy for a hovering helicopter
energy-conservation
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What you say is correct in principle, but ignores the important fact that practical car engines are horribly inefficient, and their effeciency changes quite a bit over the range of speed and power required to move the car. Note that this is the point of transmissions. At best they don't loose any power, but they make the overall process more efficient by allowing the gasoline engine to operate at a more efficient point.
In one way, you can look at a hybrid as having a wide-ranging finely adjustable transmission, but there's more to it than that. The efficiency of a gasoline engine is in part related to what fraction of peak power it must put out. If the gas engine is the only mechanical output in the car, then it must be sized to supply peak power. However, most of the time much less than peak power is needed, so the engine often runs at a inefficient point.
With a electric motor available to fill in the when peak power is demanded, the gas engine can be sized smaller and it is easier to make it more efficient over most of the normal operating range. It also allows for the option of not using the gas engine at all at very low power levels where it would be very inefficient. Instead it can effectively be run in bursts of more efficient operation. For example, if the gas engine is 3% efficient at 500 W, but 6% efficient at 1 kW, then you're better off running it at 1 kW half the time instead of at 500 W all the time. With a hybrid, you have this option. With just a gas engine, it's stuck having to produce whatever power is demanded at the moment, regardless of how efficient that is.
I have a Honda Civic hybrid, and I can tell you this stuff really works. I routinely get 50 miles/gallon minimum on the highway, often substantially more. The engine is physically small for the size car, and it has been specially designed to be easily shut down and restarted. Going down a hill, even at highway speeds, the engine often turns off. If the hill is steep enough, the motor is run as a generator and charges the battery. When I get to the bottom of the hill, I can see that for a little while the control system uses the electric motor to keep the car going at the set speed (this is all with cruise control engaged), then eventually gives up and switches on the gas engine. I can feel a slight klunk when that happens, and the charge indicator goes abruptly from discharge to charge.
The sun is turning hydrogen into helium, which produces energy from mass. This energy is ultimately radiated as an enormous amount of light: about $1370\mathrm{W}/\mathrm{m}^2$ reaches the top of the Earth's atmosphere.
Solar panels take some of this light energy and make electricity from it (note they aren't using what we would think of as heat: they need shorter-wavelength light).
Wind turbines work indirectly: some of the incoming energy from the sun ends up heating the atmosphere: differential heating causes movement of air, which is wind. This movement has kinetic energy, and wind turbines extract energy from this. The processes involved in turning sunlight into wind, while straightforward in principle, are actually extremely complex in detail.
So both of these systems rely on energy coming from the Sun. Of course, so does a car: in the case of a car, light energy from the sun gets captured by photosynthesis and turned, ultimately, into oil. That captured energy is then used, eventually, to run the car.
So the only place, in this entire system, where energy is 'not conserved' is in the Sun. And that's because, of course, energy and mass are really the same thing, and this is what is conserved. So the Sun is (very slowly) turning some of its mass into energy.
Finally note that the law of conservation of energy can't be proved: physical laws can only be disproved: this is an important point to understand. What you should ask, rather, is how do wind turbines & solar panels agree with energy conservation, or how do they fail to disprove it, not how do they prove it.
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The kinetic energy is being turned into the kinetic energy of the air, and also energy dissipated as heat and sound by frictional forces internal to the helicopter. By constantly using your rotors to push air to provide lift, you need to constantly give the air around you kinetic energy. Imagine you have a helicopter in vacuum and it has no frictional forces. There would be no energy (no fuel) needed to maintain the rotor's motion - once you got it in motion it would just keep spinning at a constant rate. Now imagine you're in the air and your helicopter has internal friction. Now you need fuel and energy to maintain your rotor's motion because otherwise your rotors would stop spinning eventually.