How where heavier elements such as carbon and oxygen formed?

Depending on the size of the star, the core compression after each elemental fuel type is used up may or may not produce a temperature which allows the next fusion scenario to proceed.

For Deuterium, it is about 1 million K; for Hydrogen it is about 4 million K and for Helium it is 100 million K. These are big jumps.

Our Sun will burn Hydrogen and then Helium. It does not have enough mass to compress the core and heat it to burn Carbon.

In Stars above 8–10 Solar masses, they will be able to run through all the elements available for fusion until it gets halted by Iron, as fusing Iron requires more energy than it gives out.

Hydrogen fusion produces mainly Helium during the process, if the proton-proton method is prevalent, or also some Carbon and Oxygen and Nitrogen if the CNO path is used