Hi Alexander, as Mark says pop along to the intro section - we are nosy and want to know all about you Lol Seriously apart from folks wanting to welcome you, it's the place people look to see what type of ABing you are into, what your AB goals are, what equipment you have, and roughly where you are in the world to advise where to get stuff etc. It's then much easier for people to give you relevant advice, as there's a few different areas of airbrushing.
Unfortunately the kits are not usually any good. Starting with the airbrush, for artwork you will need a double action airbrush. Single action is really only good for covering large areas. It is probably also very cheaply made, which means cheap materials, and poor manufacturing, usually resulting in tools that never work, work intermittently, and don't last five minutes. It is possible that the nozzle is blocked, or that soft metal parts have been damaged. If a good clean doesn't solve it I'm afraid it could just be a bad brush. If you are not using the correct paint the nozzle will block very easily. Are you getting bubbles in the paint/water?
I'm also betting the compressor is a small tankless job, that runs all the time? This won't be any good either. Without a tank it will pulse air so the airbrush won't get a consistent supply or be able to work properly. Also running constantly it is likely to overheat, and then stop.
When folks know more details about what you want to do, they will be able to give more specific advice that will suit your needs. But as you are finding out cheap airbrushes are usually no good. Even if you get one that works, it can slow down learning, and give poorer results because it will never perform like a branded brush.