iPhone 5: Lagging and hanging issues with iOS 9

I have an iPhone 5 (16 GB) and have been using it for the past 2 years. Recently I updated the OS to iOS 9 (v 9.0.2). Since then i have been facing a lot of lag while using the phone. Also, sometimes the phone becomes really slow (for instance, while typing a message, I keep on typing but no text comes up on the screen, and suddenly after ~8-10secs the text appears on screen) or simply stops responding to any input (screen touch / button press).

Similar issues weren't there with the earlier version of iOS.

I searched for troubleshooting methods for this on the internet and found that iOS 9 has incorporated a lot of visual additions that are seen to slow down the performance of the older iPhone models. For this, I reduced the motion (Settings>Accessibility>Reduce motion>ON) and the transparency (Settings>Accessibility>Reduce Transparency>ON). Also, I freed up space on my device and now have more than a gigabyte free space available.

Even after doing the above steps, I'm still facing the same problem in my phone.

Is there anyone who was facing a similar problem on the same device and was able to rectify it?

Any help will be much appreciated. Thanks!