I am using the Content Link functionality of the Content Editor web part to
pull in an HTML snippet from another server.

It seems that the content is getting cached. The HTML snippet was updated,
but the old content is still showing up in the Content Editor Web Part. When
I click "Test Link", I see the new content, but the page shows the old.

Is there any way to control (turn off) this caching behavior?

I did make sure the page that includes the Web Part is being refreshed by
adding some parameters onto the query string.