Is overpopulation real or legend?

The folks at Kurzgesagt are decidedly in the camp that believes in the grand scheme of things, the "legend of overpopulation" is not a cause for concern. Their case is based on a four-stage theory of demographic transition in a country or region:

Population Education has more on this model:

In Stage 1, which applied to most of the world before the Industrial Revolution, both birth rates and death rates are high. As a result, population size remains fairly constant but can have major swings with events such as wars or pandemics.
In Stage 2, the introduction of modern medicine lowers death rates, especially among children, while birth rates remain high; the result is rapid population growth. Many of the least developed countries today are in Stage 2.
In Stage 3, birth rates gradually decrease, usually as a result of improved economic conditions, an increase in women’s status, and access to contraception. Population growth continues, but at a lower rate. Most developing countries are in Stage 3.
In Stage 4, birth and death rates are both low, stabilizing the population. These countries tend to have stronger economies, higher levels of education, better healthcare, a higher proportion of working women, and a fertility rate hovering around two children per woman. Most developed countries are in Stage 4.
A possible Stage 5 would include countries in which fertility rates have fallen significantly below replacement level (2 children) and the elderly population is greater than the youthful population.

Overpopulation – The Human Explosion Explained (YouTube / Kurzgesagt)