Air is around 21% Oxygen. The bulk of air is non-flammable Nitrogen. Wouldn't surrounding the jet fuel with only O2 create more efficient and clean combustion?
Vehicles which carry their oxidizer with them are called rockets. It is a good thing that air is not pure $O_2$, because the nitrogen, being employed as a process gas, helps to get useful work from the combustion process. By being heated and accelerated in the combustor, it does the main work of driving the turbine and turning heat into mechanical work. Engines would be even cleaner if less oxygen is around, because already at 21% $O_2$ they run at fuel-to-air ratios well below the stoichiometric optimum.
If you consider the amount of air flowing through the core of a modern turbofan, you will see that the amount of oxygen which needs to be available will dwarf the amount of fuel, and most of it would not burn but would replace nitrogen in its role as the process gas. It could not be carried with the airplane nor isolated from surrounding air if any meaningful payload and range should remain.
For some background on the combustion process, please see this answer.
Then the weight of the oxidizer would weight down the aircraft. For each kilogram of pure octane (one of the components of jet fuel) you would need around 1.7 kilogram of pure O2. In other words it would double the weight of the fuel you have to bring.
Also bringing your own oxidizer is what happens with rockets, it does provide higher thrust and can work where no oxygen is available in the environment like in space.
As Ratchet freak has already mentioned, carrying oxygen would weigh an aircraft down.
As Peter has mentioned, a jet engine pumps air and burns some of the oxygen, not the full 21% available. Often it burns less than half of the available oxygen.
The reason for this is that the "adiabatic" (that means, assuming no heat transfer) flame temperature for complete combustion of all the oxygen in air is about 2000C. Add to that the fact that the compressor can preheat the incoming air up to 500C in some cases, and you would be looking at a temperature at the exit of the combustion chamber of 2500C.
A jet engine that could use all the oxygen in the incoming air would be more efficient, in accordance with the principles of the Carnot cycle, but unless built of "unobtanium" it would melt and fall apart.
A jet engine needs a turbine to do two things. The first is to drive its own compressor. The second is to drive a load (either a propellor or fan.) The turbine+prop/fan is needed because the exit velocity of the exhaust gases would be too high to be used efficiently for generating thrust. Hence putting a turbine in the exhaust path is unavoidable
A rocket has no turbine in the exhaust path, and as a result can run at the extremely high temperatures generated by burning oxygen. It also has extremely high exhaust exit velocities, which are hopelessly inefficient at the kind of vehicle velocities required for general aviation, but are excellent for space travel, where the more energy you can impart to each ton of propellant, the less propellant you have to carry. Also most rockets to date have been single use, unlike jet engines.
So we are stuck with jet engines which are unable to use all the oxygen in the fuel, due to materials issues. One way of boosting the power of a jet engine (but not the efficiency) is to use an afterburner. This is effectively a second combustion chamber, behind the turbine, where more fuel can be burnt. This creates a huge amount more energy, but comparatively little extra thrust. Remember kinetic energy=0.5mv^2 while momentum=mv. You need to match the exhaust velocity to the vehicle velocity for efficient thrust generation, and an afterburner does not do this, as its purpose is not efficiency, just a quick and dirty power boost.
There are some designs on the drawing board for the very large gas turbines in power stations, where the gas from the turbine, having been cooled by expansion, is sent to a second combustion chamber and then through a second turbine, to enable a higher average heat-addition temperature and improve efficiency. However the benefits are small and as far as I know it has not been tried yet. I suspect it will always be too complex for aviation.
More viable than oxygen would be water injection. Water/steam is now being used for cooling of industrial gas turbines, which reduces the amount of air that must be pumped by the compressor, improving efficiency. However, you would still have to carry it. Another option (also borrowed from industrial gas turbines) would be to put a heat exchanger in the exhaust gas flow, in order to generate steam for a steam-turbine driven propellor or fan. However, I don't expect to see these tried until all the oil in the world has nearly run out!
Basically, reaction engines are more energy efficient the larger the amount of propellant in relation to fuel is. With a jet engine the propellant is air which happens to already contain more oxygen than is needed to burn the fuel. In theory this might change for a very fast aircraft requiring very fast exhaust velocity, but the engine design would have to change too much for it to be relevant to the question.
The general benefit from more efficient burning (barring some specific performance requirement) would also be better fuel efficiency. Getting better fuel efficiency at the expense of carrying additional oxygen is unlikely to be useful.
I actually toyed with a concept for an engine similar to your idea, but, as mentioned by others, an engine that carries its own oxidizer is basically a rocket, so my concept was really a rocket-powered-bladeless-fan with characteristics similar to jet engines not an actual jet engine. So even approaching it from the opposite direction the answer is "The design would have to change too much."
People have actually tested engines based on using conflagration or detonation instead of combustion for more efficient burning of fuel. See Pulse Detonation Engine for example. These still fall in the category of "The design would have to change too much." Also note that they have not replaced jet engines despite the concept having been known for decades. Wikipedia has lots of articles on rare or experimental engines. You might find something close to your idea there.