I've been told that GR and QM are not compatible, is there an intuitive reason/thought experiment which demonstrates the issue? (Or one of the issues?)
The simple thought experiments are not related to the naive reason people give for the incompatibility--- people say that they are incompatible because of renormalizability. This is not easy to explain, and it is in fact false, because N=8 supergravity is with some scientific confidence renormalizable. But it's not a good theory.
The reason for the incompatibility is the behavior of black holes--- the fact that they have an entropy which scales as the area, not the volume. In quantum field theory, the fields near the horizon have infinite entropy, as shown by 't Hooft. So you need a different kind of quantum theory, one which is nonlocal enough to allow black holes to have entropy which goes as area.
This is the only incompatibility, since string theory is exactly such a quantum theory, and it is a consistent theory of gravity. So there is no further incompatibility left.