Bribery is also officially illegal in China and many other developing countries and people have been put to death for corruption.
It's pretty ridiculous to say that bribery in the West gets uncovered because it's illegal. Some of it comes to light, sometimes it's swept aside or a scapegoat gets the brunt of the blame and no doubt there's plenty that also goes unnoticed just like many other crimes.
Just because the US currently manufactures for more dollars than any single country doesn't necessarily mean that the West has shifted a lot of it in developing countries. Vehicles, pharma and raw materials will likely stay in place.
For many companies, the whole point is to escape the rules and regulations and take advantage of the fact that corruption allows for this in some countries whereas in most western countries, it is now too difficult or costly to operate. Certainly lower wages is usually the biggest factor. Corruption is about opportunity and risks, not about race or nationality. No one is inherently morally superior, the environment often shapes the behavior.
I've seen many people behave in very different ways when put in vastly different environments and morality becomes a very shifty concept.