How to use Sentry in an SDK that's installed on other sites?

I’m building a JavaScript SDK and I want to log errors thrown in my SDK as well as log events in it. I decided to install @sentry/browser from npm and it works great (barring the bloat in package size).

However, I’m also getting the errors of the websites that install my SDK, which I don’t want to see. Is there a way to limit Sentry so that I only see the errors in my SDK alone?