Contrary to the impression you get from the popular descriptions gravitons don't contradict gravity being a curvature of spacetime. Graviton arise as an expected quantum behaviour of ripples of the gravitational field - gravitational waves of very small intensity.
The classical nonlinear equations can be solved exactly only in few cases. In the quantum theory it's even worse. Usually we can only solve very special field theories known as free field theories. However we can approximate interacting QFTs in certain regimes by the free QFTs. Basically you construct a model of tiny ripples about some nice background. These tiny ripples are known as quantum particles.
It does not really matter what is the nature of the full theory and various ripples of various stuff behave in a similar way. E.g. tiny sound waves also behave as quasiparticles known as phonons.
Then you take into account that your model is not a free QFT by introducing the interactions between those particles. This gives rise to the "virtual particle exchange" picture which is just a description of non-propagating field configurations in terms of our ripples.
But doing this we don't throw away what we know about the full theory. This particle exchange is just an artifact of our description in terms of ripples. But more often than not this is the only way we can do any calculations in the quantum regime.