4 digits 7s display + 74HC595

I don't see how. You have all four cathodes turned all the time

digitalWrite(digitOne, HIGH);
digitalWrite(digitTwo, HIGH);
digitalWrite(digitThree, HIGH);
digitalWrite(digitFour, HIGH);

I see you did manage to incorporate an array to look up the font, that's a good sign

shiftOut(dataPin, clockPin, MSBFIRST, digit[i]);