Safari update on iPhone

There is no way to update Safari by itself. The only way it would receive an update is through an iOS update. However, your iPhone appears to be on iOS 10 -- and that is the latest version available for iPhone 5c. Nonetheless, you can try seeing if there's an iOS update available just in case you need to update iOS 10 further.

In addition to MSAppleIX’s comments, bear in mind the iPhone 5C was released by Apple in 2013.

So how can you update your Safari browser? By purchasing a newer iPhone.