If measurement fixes the state of a quantum system, how do we know that superposition exists? My very limited understanding of quantum mechanics cannot make sense of the superposition phenomena. It would seem to me like if a measurement makes the wave function of a quantum system collapse, nobody has ever experienced superposition?
Edit, to be more precise:

*

*In order to observe a physical phenomena, one has to undergo the process of "measuring" it

*When a quantum system is in a superposition of states, the act of measuring it makes the wave function collapse before the measurement, fixing it to one of the possible states.

Therefore, any particle we ever observe is fixed in one state. I know this must be wrong, but why?
 A: 
How do we know superposition exists?

We can do some math to predict the behavior of how certain "quantum" things behave. One of the most popular interpretations is that these things can exist in quantum "superpositions" in which the quantum thing is simultaneously in multiple states at once. It's not the only interpretation so it's not necessary to believe it.
Now what exactly is a "superpositon state"? Well, essentially the mathmatical model assigns independent complex numbers to each possiblity. So for example if I flip a coin, each possiblity receives its own number, so heads receives a complex number and tails receives a complex number. Sometimes, when different quantum states interact, the outcome of the resultant output state is a sum of these assigned complex numbers - which we interpret as an "interference between the different possibilities." It is this interference between different possibilities that we interpret as "superposition." And often this effect looks like an interference between waves (which you can see in the double slit experiment, for example).


*

*In order to observe a physical phenomena, one has to undergo the
process of "measuring" it

*When a quantum system is in a superposition of states, the act of measuring it makes the wave function collapse before the measurement,
fixing it to one of the possible states.  Therefore, any particle we
ever observe is fixed in
one state. I know this must be wrong, but why?


We only ever measure something in a fixed particular state (so you are correct here), but what is interesting is which particular state we observed. To predict the correct outcomes, we use a model that requires interference between the numbers assigned to each possibility. We then conclude that the model is correct and that when a state is not measured, all of these numbers assigned to each possibility actually exist and that is what superposition is.
A: All other answers to this question do an excellent job of delving into the math behind it.
I would like to address the kind of "philosophy of science" aspect of it , because i think that is what the OP was curious about as well.
His question seemed to boil down to , "how do we know superposition exists if we can never observe it" .
The answer to this is simply that superposition is an interpretation or model of what is happening. This model gives a very accurate predictions that match with observations. This is why we believe in it.
That is true for every other established model in science. We believe  in the model of quarks, electrons and other sub atomic particles because that model provides predictions that are consistent with observations.  Same goes for electric or magnetic fields etc. or ANY other model we have in physics.
Things are not more "real" just because we can see or touch them . When we are seeing or touching something, those are also experimental outcomes. Just like the path of a particle in a cloud chamber or the observed effects of decay products etc. are experimental outcomes
A: Many other answers explain the math, but I think the crucial point is that your understanding is wrong in the second point:

When a quantum system is in a superposition of states, the act of measuring it makes the wave function collapse before the measurement, fixing it to one of the possible states.

The wave function collapse does not happen before the measurement, it is a result of the measurement. If your system is in a state $|\psi\rangle$, you can thus measure any property of $|\psi\rangle$ you would like -- just be aware that the system state will not be $|\psi\rangle$ after the measurement.
For example, if you prepare a qubit in the superposition $|\psi\rangle = \frac{1}{\sqrt 2} ( |0\rangle + |1\rangle )$ of its ground and excited states and measure its excitation level, you will receive "0" or "1" with 50% probability each. Therefore, if you are able to prepare the qubit in that state multiple times, you will get a different measurement result each time, indicating the superposition.
(Note that this simple experiment cannot distinguish a superposition state and a classically mixed state, but that is a more advanced topic.)
A: The point is that if you fix the position, say, of a particle, you are observing it in a superposition of momentum states. Conversely, if you observe a particle having a specific momentum, ie in a specific momentum state, you are observing it in a superposition of position states.
It is a very elegant characteristic of quantum mechanics that an eigenfunction of one observable operator can be expanded mathematically as a superposition of the eigenfunctions of another.
A: Superposition of states is a very mathematical concept which has very less physical meaning.
what superposition says is If there are two states which satisfy the Schrodinger equation, then a linear combination of them would also satisfy the Schrodinger equation. Blindly converting this to make physical sense : If there are two states in which a particle can be, then the particle exist in both states at once (statement 1).
This is as good as saying the wave $f(x) = 3sin(x) + sin(4x)$ looks like this : fig1 and not like this : fig2


But the concept of superposition is very much necessary to explain the result of some experiments. A crazy thing happens when we try to measure any observable property of a quantum particle (the property should be different for both the states. Like say the wavelength of $f(x)$ ). Experiments done show that if we measure any property of the particle, we only get one result corresponding to any one of the state. And it happens randomly everytime. As in our example, if we measure the wavelength of f(x) it randomly gives $2\pi$ or $\pi/2$ under repeated measurements. (But more often it will give $2\pi$ since it has a greater amplitude.)
So how we know superposition exist? We know because of the fact that when we repeatedly measure the state we get all the possible states. But one at a time.
Let me add that the idea of collapse and superposition of wave functions exist because those are the only way we are able to explain the results of experiments. Better theories or corrections are necessary to understand the laws of nature.
