How can we disprove quantum oddities like superposition and entanglement?

We do not necessarily disprove them. Quantum mechanics has reached scales so small that you can only theorize what happens; that is why you have the Heisenberg uncertainty principle and the electron cloud of probabilities. But take the double-slit experiment: every animation shows a single photon or electron passing through and mysteriously interfering with itself. In reality there are thousands of photons going toward the slits, some a fraction of a second ahead of others. So what they treat as one electron interfering with itself may actually be two electrons, each with its own de Broglie wavelength, producing the interference pattern. Since you cannot shoot or even see a single photon or electron at a time, we are not even sure superposition and entanglement are correct; they may be claiming something that is beyond their instruments.