Mac Os 10.7.5 and Safari 6.1.6

You need to upgrade OS X to upgrade Safari.

Upgrading to El Capitan or Sierra

You can upgrade to El Capitan or Sierra from Lion or directly from Snow Leopard. El Capitan or Sierra can be downloaded from the Mac App Store for FREE.

To upgrade to El Capitan or Sierra you must have Snow Leopard 10.6.8 or Lion installed. Download El Capitan or Sierra from the App Store. Sign in using your Apple ID. El Capitan or Sierra is free. The file is quite large, over 5 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.

The official list of Macs compatible with macOS Sierra:

2010 or Later

MacBook Air

MacBook Pro

Mac mini

Mac Pro

Late 2009 or Later

MacBook

iMac

Macs that can be upgraded to OS X El Capitan

1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later

2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later

3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later

4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later

5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later

6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later

7. Xserve (Early 2009) - Model Identifier 3,1 or later

To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.

If your computer cannot be upgraded to Sierra or El Capitan, then you are stuck with Lion.

Are my applications compatible?

See App Compatibility Table - RoaringApps.

those pre-2008 Mac Pros do not have 64-bit firmware, so are limited to 10.7.5 as the latest officially-supported version of MacOS. Since the version of Safari is based on the version of MacOS, you are already running the latest officially-supported version of Safari that can run on that computer.

You can buy yourself some time by running the appropriate version of Firefox that runs under 10.7.5, but I think only about half a year remains before it will no longer be updated.

There are some hobbyist hacks that can install a version of 64-bit kernel on top of (and dependent on) the 32-bit kernel in your machine. It is a little complicated to set up, and will be good for One version of MacOS only (if you decide to run a different version, you need to put in the hacks again). If you are not "handy" it is not likely to be a solution for you.

Get the latest version of Safari for your Mac - Apple Support

 menu > About this Mac

what is the top line that says what model Mac you are running? (please do not post your serial number)