  It is still a matter of intense debate how supermassive black holes (SMBH)
grow, and the role played by feedback from active galactic nuclei (AGN) in the
co-evolution of SMBHs and galaxies. To test the coevolution proposed by
theoretical models, we compile a large AGN sample of 5639 X-ray detected AGN,
over a wide redshift range, spanning nearly three orders of magnitude in X-ray
luminosity. The AGN have been detected in the {\it{COSMOS-Legacy}}, the Bo$\rm
\ddot{o}$tes, the XMM-{\it{XXL}} and the eFEDS fields. Using the specific star
formation rate estimates, we split the AGN host galaxies into star forming
(SF), starburst (SB) and quiescent (Q). Our results show that the AGN accretion
is increased in SB systems compared to SF and Q. Our analysis reveals a mild
increase of L$_X$ with M$_*$. The L$_X$/SFR ratio has a weak dependence on
M$_*$, and at fixed M$_*$ it is highest in Q systems. The latter trend is
mostly driven by the significant drop in SFR in the Q state. The measured
strong variations in SFR from the SB/SF to Q mirror those predicted in merger
models with AGN feedback. However, the observed mild variations in L$_X$ are at
variance with the same models. We also study the evolution of SFR for a galaxy
control sample and found that it is very similar to that of X-ray AGN. This
suggests that either AGN play a minor role in the star formation quenching, or
the relative timescales of the two processes are different.
