Can I get a 'modern' browser for Mac OS X (10.7.5) without upgrading to OS X El Capitan?
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.