Here are the list security vulnerabilities found in Firefox over time and each of them have been fixed in their corresponding higher versions.
How to know if your web browser (or any software for that matter) is unsafe?
If you are not using the latest version of that software, you are not safe.
I am surprised to learn that, why am I easily allowed to browse the web if it is unsafe?
Visit the Firefox website from your Firefox 47 and you will probably be notified on the top of the site that you need to update your browser for safe and fast browsing.
For safe browsing, you need to install the latest version of Firefox, install some useful browser extensions like HTTPS Everywhere, Privacy Badger, NoScript. You also need to have some conscience.