Since synchrotron radiation is created when charged particles are radially accelerated and electrons are definitely orbiting a nucleus (assuming a Bohr model), electron should then logically emit synchrotron radiation. However, if it does, then it lose energy and would unfortunately spiral into the nucleus, which we know doesn't happens. So, is it that atoms doesn't produce synchrotron radiation or other mechanisms are compensating the synchrotron radiation effect?
The Bohr model is wrong. You can get a lot closer with the Schröodinger picture, and when you do that you find the orbitals which are not the same thing as orbits: they are quantum states not classical paths.
The bound states (orbitals) of atoms are not time-dependent, so they don't radiate.
Well, that's absolutely true for the ground states. The non-ground states do spontaneously couple to the photon-field but they do so more or less discretely.
Excited states of atoms gradually goes into less excited states and photons, eventually stopping at the ground of state of an atom. Such a cascade of spontaneous emission is a direct continuation of the synchrotron radiation as you shrink the orbit of an electron from synchrotron scale to atomic scale. There is not much fundamentally different at the atomic scale, except for the stability of the ground state and discreteness of radiation produced.