When you say turn it off, do you mean pressing the power button for ~1 second until it goes off?
If so, you are cutting off power to the eTrex 20. This is similar to removing the batteries from the unit. It will not track in this mode.
Some of the Garmin models (e.g. Oregon) has a "screen off but still powered on" mode where it will continue to track. I took a peek at the eTrex 20 manual, there doesn't seem to be such a mode.
To reduce battery consumption due to the screen being on, you can adjust the backlight timeout as well as screen brightness.