Why can't physicists make gravity quantum?

Because the two best descriptions of nature disagree about what space and time even are. Three of the four fundamental interactions, electromagnetic, strong and weak, are described within quantum mechanics and quantum field theory, leaving gravity as the only interaction not fully accommodated. Quantum gravity is the search for a theory that fixes this, which we need wherever neither gravitational nor quantum effects can be ignored: near black holes, and moments after the Big Bang.

Einstein's general relativity models gravity as curvature of spacetime. In John Wheeler's slogan, spacetime tells matter how to move, and matter tells spacetime how to curve. Quantum field theory, on the other hand, is typically formulated in the flat spacetime of special relativity. If you try to treat gravity as just another quantum field, the resulting theory is not renormalizable: in perturbation theory it needs infinitely many independent parameters, and since infinite experiments are impossible, it has been argued that there is no meaningful physical theory.

There is a deeper clash about time. In quantum theories, time is an independent background through which states evolve. In general relativity, time is a dynamical variable. Quantum mechanics has depended since its inception on a fixed background, whereas general relativity has none.

Candidates exist. String theory replaces point particles with strings, one mode of which is always the graviton, at the price of extra dimensions. Loop quantum gravity finds that area and volume are quantized, so spacetime has a granular structure at the Planck scale. But direct effects are thought to appear only near the Planck scale, around 10⁻³⁵ metres, so physicists lack experimental data to distinguish the competing theories.