I understand why hydrogen and helium are the most common elements in the universe. I also understand iron, because that's where stellar fusion stops. But why are (specifically) oxygen, carbon and neon more common than iron? I would expect iron to either outrank them, or some steady flow of atomic nnumbers to lead up to iron. These three, and the scattershot of seemingly random elements trailing behind iron, seem a very odd mix!
This is a very broad question: its answer involves the full details of stellar evolution, Galactic chemical evolution and nuclear physics.
I'll limit myself to the following observations:
The elements you mention (actually are all "islands of nuclear stability". Whilst their binding energy per nucleon is not as high as that of iron, it is a little higher than elements immediately around them in the periodic table.
Nuclear fusion reactions are exothermic (up to iron), but they require energy to initiate them (in a similar way that you need to get coal hot before it will burn). This ignition becomes more difficult the higher the atomic number (the number of protons) in a nucleus. This means that it is not a given that nuclear reactions will simply proceed to the element with the highest binding energy. If there is a way to prevent the material in a star reaching this ignition temperature, then the reactions will not proceed through to iron.
This means that even in a very massive star, it is only the core where burning runs to completion in the iron peak elements. The majority of the star is still hydrogen and helium, with the layers outside the core containing the remnants of burning of lighter elements. It is these products that get blown into space in a supernova and mostly consists of products of helium burning (C/O) and alpha capture onto these elements (Ne, Mg, Si, S, Ar, Ca), with each being slightly harder to produce than the last and accounting basically for their relative abundances. Most iron would end up trapped in the neutron star or black hole remnant.
Iron is mostly produced following thermonuclear detonation of C/O white dwarfs - a completely different production path. Hence the abundance of Fe is decoupled from things like Oxygen, and more dependent on the rate of type Ia supernovae, governed by binary interactions and statistics than the details of nuclear physics.
Carbon and Nitrogen are mainly produced in lower mass stars that never become supernovae and where conditions never reach ignition temperatures to produce heavier elements by fusion. There are many more of these than stars which end as supernovae.
So, the relative abundances of the elements is a very complicated issue that has no simple answer, but where the broad details are well understood in terms of the processes I mention above.