Hi,
I want to share my troubleshooting steps about a topic rather then asking a question. Maybe future visitors may benefit from this thread because I was not able to find enough resource about this problem.
I am an inexperinced user. I bought an Ardunio Nano clone with LCD Keypad Shield and a I2C backpack for hobby purposes.
I hooked up everything according to this article but something was wrong. The screen was going off immediately after a brief light up when I connect it to PC.
I wrote some simple "Hello World!" code with this LiquidCrystal_I2C library.
The result was the same. It was displaying the text for a brief moment and turning off.
I thought it was a power issue. I bought a Nano shield and powered it with a 9V 2A adapter. No change... All I was able to think either it is a faulty hardware or a bad clone.
Then, I found this article. Was it only a backlight lighting issue all the time? I powered my setup and checked the screen with my phone's flashlight. Yes, the text was there! But I was afraid to do that hack because I neither have the right tools nor skills. Also this mod introduces the need of backlight brightness control via code -at least as far as I understand (EDIT: Apparently not).
I decided to pull the cable on pin 10 while it was still powered. Suddenly the screen lighted up and stayed on with no text. I seeded the cable again and reset the code. It was still on with text this time 
Whenever I restart the system, the screen is booting perfectly after all.
Now, all my concern is, if I connect another device and the communication is interrupted, will the backlight go off again? I will try this scenario as soon as possible. I am also wondering your opinions and comments.
Thanks for your time and reading!