Few months back, Instagram launched the dreadful Activity status feature. Basically, it shows the last active time of people on Instagram with whom you have had direct conversation. If that wasn’t enough, recently they introduced the online status indicator. When a person is online, you will see a green dot next to their name in Direct Messages (DM).
Here are the steps to turn them off.
Step 1: Go to your profile and tap the three-dot icon or the gear icon.

Step 2: Scroll down and tap on Activity status. On the next screen, disable Show activity status. This will turn off activity status and green dot both.

