My ipod t is 4th generation.Not able to download any compatible Apps

If you already purchase apps for this iPod previously there is a way to get all of these apps back.

Go into the iOS App Store app and under the Purchases tab/section/button in the lower menu.

Then, try tapping the download “cloud” icon next to the app/s, in this location, to see if the small window pop-up prompt that states that there is an older version of the app that is compatible with your device. Would you like to download this version now?

Tap the OK button just below the prompt to start the downloading of the older app version to your iPod Touch.

Best of Luck to You!

The 4th generation iPod touch cannot go beyond 6.1.6

try to update the latest version.