Can I get a 'modern' browser for Mac OS X (10.7.5) without upgrading to OS X El Capitan? I am using Safari (6.1.6) for all my web use but certain sites I use often are telling me my browser is out of date. I gather Safari cannot be upgraded to a more modern status.  

Having read the often less than complimentary reviews of 'El Capitan' which I assume would be the next logical step, I wonder if I can keep my OS X version but use another more up to date browser.
You can currently use Chrome, the world's most popular browser, but not for long. Google are dropping support for OS X versions before 10.9 from April of this year. A Chrome version you download now will keep working, of course, but it won't get updates.

However, it's unsafe to use the Internet from OS X 10.7.5, regardless of which browser you use. It's not received any security updates for years, so it's full of unpatched known vulnerabilities.