This is one of the funny things of chemistry. If multiple reactions can occur at the same time, then they will occur at the same time. At high-school pupils learn all kinds of chemical reactions with perfect stoichiometry, but in real life, things are much more complex and there hardly is any reaction, which runs quantitatively with a certain stoichiometry.
So, if you have sufficient voltage available, then all reactions can occur at the same time and this is what you observe. Besides that, formation of oxygen becomes easier and easier during the electrolysis. At the cathode, OH(-) ions are formed and the reaction 4OH(-) --> 2H2O + O2 + 2e is much more facile than the formation of oxygen from water or the formation of chlorine. So, over time, OH(-) will be the main supplier of oxygen at the anode.
But even if there were no OH(-), oxygen would still be formed, simply because it can be formed.