Is Using Streaming Websites Like Putlocker or Gostream Illegal in the USA?

Yes. It’s illegal. ‘Streaming’ is downloading - your computer doesn’t keep the video file long-term, but you are still downloading a file you have no legal right to.

But ignoring that, there's another reason to stay away from these sites. Video streaming sites cost money to run, and many get money from hosting ads. Because their content is sketchy, they can't get ads from good places, so they have to get ads from sketchy places that are often infested with malware. And because more clicks = more money, they make it very easy to accidentally click.

A good adblocker and script blocker helps, but it's still not recommended to visit sketchy websites.

Streaming might be downloaded. (Or more relevantly, it might be making an unauthorized copy.)

This is a legal gray area, because the definition of “copy” requires it to be nontransitory. So, whether streaming is illegal or not is going to depend on the technical implementation and a judge/jury’s familiarity with computers.

So, streaming might be illegal.

But, if you’re concerned about the consequences of streaming video, you should keep in mind that if you’re sued for copyright infringement, that you’re likely going to want to settle because it would cost a small fortune to get a ruling on the merits of your question.