How does an organ system turn into a organism?

Organisms are the product of evolution. Organisms did not evolve from organ systems, rather, organisms have evolved organ systems to accomplish various specific biological functions.

However, if we go back farther, to the evolution of the cell, we find that there are organelles, the mitochondria, which have their own genetics, and which almost certainly evolved originally as independent organisms before entering into a symbiotic relationship with cells. It is speculated that perhaps all the organelles of a cell evolved separately. That is the closest we can come, to an organ system that turns into an organism, although it is really an organelle system.