If jets carried pure O2 to use for combustion, would they be more efficient?
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.