As I said, I already tried the approach of sending a pulse to the strobe [enable pin], to retrieve the second nibble.
I'm sorry, but that is not what you said.
Oh, I believe that is what I said, from two or three posts above:
I tried sending a pulse to the strobe (enable) to get the next bits, but that caused undesired side effects, like text randomly disappearing from the screen, which leads me to believe that approach is wrong.
Since you have provided almost no information about your program I did the best with what you did provide.
I need to read from an HD44780-based LCD screen in 4 bit mode. What else do you need to know in order to provide an algorithm?