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We study the connection between active galactic nuclei (AGN) and their host galaxies through cosmic time in the large-scale cosmological IllustrisTNG simulations. We first compare BH properties, i.e. the hard X-ray BH luminosity function, AGN galaxy occupation fraction, and distribution of Eddington ratios, to available observational constraints. The simulations produce a population of BHs in good agreement with observations, but we note an excess of faint AGN in hard X-ray (⁠|$L_{\rm x}\sim 10^{43-44}\, \rm erg/s$|⁠), and a lower number of bright AGN (⁠|$L_{\rm x}\gt 10^{44} \, \rm erg/s$|⁠), a conclusion that varies quantitatively but not qualitatively with BH luminosity estimation method. The lower Eddington ratios of the |$10^{9}\, \rm M_{\odot }$| BHs compared to observations suggest that AGN feedback may be too efficient in this regime. We study galaxy star formation activity and structural properties, and design sample-dependent criteria to identify different galaxy types (star-forming/quiescent, extended/compact) that we apply both to the simulations and observations from the candels fields. We analyse how the simulated and observed galaxies populate the specific star formation rate – stellar mass surface density diagram. A large fraction of the |$z$|  = 0 |$M_{\star }\geqslant 10^{11}\, \rm M_{\odot }$| quiescent galaxies first experienced a compaction phase (i.e. reduction of galaxy size) while still forming stars, and then a quenching event. We measure the dependence of AGN fraction on galaxies' locations in this diagram. After correcting the simulations with a redshift and AGN luminosity-dependent model for AGN obscuration, we find good qualitative and quantitative agreement with observations. The AGN fraction is the highest among compact star-forming galaxies (⁠|$16-20{{\ \rm per\ cent}}$| at |$z$|  ∼ 1.5–2), and the lowest among compact quiescent galaxies (⁠|$6-10{{\ \rm per\ cent}}$| at |$z$|  ∼ 1.5–2). [ABSTRACT FROM AUTHOR]