Is it actually possible to shut down the Internet?

In a recent speech in South Carolina, Donald Trump said he would look into "closing up parts of the Internet" to fight ISIS. Well here's a fact that Trump may not be too happy to hear — America does not control the Internet everywhere in the world. Additionally, many American websites already censor terrorist activities.

So a hypothetical President Trump's only hope for turning off the Internet would be right here in America, and thanks to a very secretive government policy, he probably could.