Quantum fields vs. Special & General Relativity fields?

The incompatibility is related to the problem you bring up but not quite the way you have stated.  Let's start with quantum field theory which is fully compatible with special relativity.  QFT works well at describing the subatomic work and is part of the structure of what is called the Standard Model.  There are several ways to getting to QTF, probably the best known way is via second quantization.  Regardless, a funny thing happens as you try to predict the probabilities of events occurring in QFT - you get infinities arising. Those infinities are troublesome - they can signal that the model is flawed and you need to start over.  But then we employ a trick called re-normalization, where it is assumed that particle masses and electrical charges can be written as the sum of an infinitely large part and a finite part equal to the observed mass or charge, and the infinite part cancels the infinities that arise in the computations of the probabilities so all is good.  This works - quantum electrodynamics - the quantum theory of electromagnetism - uses this trick and has made predictions accurate in 1 part in 10^11 when compared to experiments.

Now all of the above occurs on a flat space-time and the only forces present are electromagnetic, weak, and strong forces.  If you want to add in a quantum model of gravity, you need to change to a curved space-time.  Here's where the problem begins.  The quanta of gravity, the graviton, can interact with itself.  In fact, in a quantum model, the gravtiton can interact an infinite number of times with itself between two points in space-time.  THe will lead to infinities in the calculations of probabilities for events.  If we try to renormalize, like in QFT, we find that there are no masses or charges where we can play the game of making an infinitely large piece and a finite piece.  There's nothing to renormalize. One reason this may occur is because unlike electric, weak, or string fields, there is no fundamental unit of mass or charge associated with gravity.  Any amount of mass curves space-time, and so does any amount of energy.  And space-time changes with the change in position of mass and energy. 

Getting to a quantum theory of gravity has been a topic of research for more than 60 years. 

Quantum field theory can be done in curved spacetime, although it is difficult. What can't be done is quantize the gravitational field, or spacetime. We don't know how to apply quantum field theory to curved space, but we can use it within curved space. 

Other than the obvious that we don’t have a quantized gravity similar to the QFT quantizations of the other fields, there are some fundamental incompatibilities between general relativity and quantum field theory (QFT).

QFT is dependent on the existence of a ‘background’ spacetime. It is not independent of that background spacetime. You can think of QFT as being defined as a field on - and dependent on - that background spacetime.

The equations of general relativity on the other hand, are independent of a background spacetime. This is a field on a field view rather than a field on a background spacetime. It is a difficult concept in general relativity to explain and is somewhat counterintuitive, but is extremely important.

So, what does the background independence mean in general relativity? The equations of general relativity are tensor equations that are independent of choice of coordinate system (reference frame). That is why you see them expressed in a form that doesn’t explicitly refer to any coordinates. 

However, these equations are also invariant under a class of active transformations called diffeomorphisms. This is just a fancy term for certain kinds of mappings from one form of spacetime to another form of the spacetime, the mappings including the tensors. So, you can change spacetime through the diffeomorphisms, yet preserve the equations of general relativity. In other words, the relationships between the tensors don’t change with a spacetime change, giving general relatively a deeply relational view. This gives general relativity a background independence not possessed by QFT. So, QFT can be fully invariant under the simple spacetimes of special relativity, but not under the general diffeomorphisms of general relativity

There are also explicit conflicts between general relativity and quantum mechanics at event horizons - so called firewall problem. General relativity requires that you see nothing locally special (a foundational principle of GR) when you cross the event horizon. Quantum mechanics predicts a massive energy firewall due to the energy released by breaking of entanglements required for information conservation and Hawking radiation.

There is a lot more but this should do ya ....

I think of more like this: in quantum theories, things happen in "jumps". The state of a system goes from S1 to S2, without anything in between. In relativity, things happen smoothly. The math allows you to define things as fine as you want, and there are always states between S1 and S2.

Also, in quantum descriptions of reality, particles can interact over any distance. In relativity, interactions are always local: things have to come very close ("touch") in order to interact.