The four countries with a higher per capita population are: El Salvador, Rwanda, Turkmenistan, and Cuba. (Omitting American Samoa, because that's a US territory).
None of those are major countries, so the statement "The US is the most punitive major country in the world" absolutely holds water.
If you want an idea of how crazy high incarceration rates are in the states, just compare the prison population per 100,000 inhabitants with its neighbors:
Canada - 85
Mexico - 174
USA - 505
China's prison population is lower, but they have "administrative detention" which isn't included.
How to lie with statistics.
This is the same for many other countries but not the case for the US. Also your first two examples applies to really small list of countries (Iran and S.A.) and their current governments are the disgrace of the world and are not really an example to take (like going too extreme and something the US shouldn't "I guess" be compared too).
For this statement to hold, it requires the most absurd definition of "punitive" imaginable.
There are countries where talking negatively about the wrong people is grounds for execution.
There are countries where theft is punished by amputation of a limb.
There are countries where not flushing a public toilet that you used, is a punishable offense.
All of these countries are much more punitive, and almost all of them have much less crime than the United States.