Overpopulation is an amplifier. Relatively solvable issues become major crises when scaled up in size because of population: epidemics, pollution, resource consumption, demand for geographic space, etc.

As societies sense this, they tend to self-regulate their populations, in part because the cost of children (schooling, medical care, housing, and so forth) increases. So you get the demographic transition model (graph courtesy of High School Geography):

A more detailed discussion of this model is here.

But the real question that remains is: will the population of the world overall regulate and reduce in sufficient time to prevent the amplifying effects of other secondary issues from causing widespread crises? Or will it take one of those crises (a good epidemic or war) to act as the population reducing agent?