To find the probability that 4 randomly illuminated bulbs form a 2x2 square on an $x$ by $x$ grid, we need to calculate:

1. The total number of ways to choose 4 bulbs out of $x^2$ bulbs.
2. The total number of ways to form a 2x2 square in an $x$ by $x$ grid.

### Total Ways to Choose 4 Bulbs
The total number of ways to choose 4 bulbs out of $x^2$ bulbs is given by the combination formula:
$$\binom{x^2}{4} = \frac{x^2!}{4!(x^2 - 4)!}$$

### Total Ways to Form a 2x2 Square
To form a 2x2 square, the top left bulb of the square can be any bulb except those in the last row or the last column. Therefore, there are $(x-1)$ options for the row and $(x-1)$ options for the column for the top left bulb of the square. So, the total number of 2x2 squares that can be formed is:
$$(x-1)(x-1) = (x-1)^2$$

### Probability
The probability $P$ that the 4 bulbs form a 2x2 square is the ratio of the number of ways to form a 2x2 square to the total number of ways to choose 4 bulbs:
$$P = \frac{(x-1)^2}{\binom{x^2}{4$$

Let's calculate this probability in terms of $x$.

The probability that 4 randomly illuminated bulbs form a 2x2 square in an $x$ by $x$ grid, in terms of $x$, is $\frac{24(x - 1)}{x^2(x + 1)(x^2 - 3)(x^2 - 2)}$.