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We investigate galaxy sizes at redshift |$z\gtrsim 6$| with the cosmological radiation-magnetohydrodynamic simulation suite thesan(-hr). These simulations simultaneously capture reionization of the large-scale intergalactic medium and resolved galaxy properties. The intrinsic sizes (⁠|$r^{\ast }_{1/2}$|⁠) of simulated galaxies increase moderately with stellar mass at |$M_{\ast } \lesssim 10^{8}{\, \rm M_\odot}$| and decrease fast at larger masses, resulting in a hump feature at |$M_{\ast }\sim 10^{8}{\, \rm M_\odot}$| that is insensitive to redshift. Low-mass galaxies are in the initial phase of size growth and are better described by a spherical shell model with feedback-driven outflows competing with the cold inflowing gas streams. In contrast, massive galaxies fit better with the disc formation model. They generally experience a phase of rapid compaction and gas depletion, likely driven by internal disc instability rather than external processes. We identify four compact quenched galaxies in the |$(95.5\, {\rm cMpc})^{3}$| volume of thesan-1 at |$z\simeq 6$| and their quenching follows reaching a characteristic stellar surface density akin to the massive compact galaxies at cosmic noon. Compared to observations, we find that the median ultraviolet effective radius (⁠|$R^{\rm UV}_{\rm eff}$|⁠) of simulated galaxies is at least three times larger than the observed ones at |$M_{\ast }\lesssim 10^{9}{\, \rm M_\odot}$| or |$M_{\rm UV}\gtrsim -20$| at |$6 \lesssim z \lesssim 10$|⁠. The population of compact galaxies (⁠|$R^{\rm UV}_{\rm eff}\lesssim 300\, {\rm pc}$|⁠) galaxies at |$M_{\ast }\sim 10^{8}{\, \rm M_\odot}$| is missing in our simulations. This inconsistency persists across many other cosmological simulations with different galaxy formation models and demonstrates the potential of using galaxy morphology to constrain physics of galaxy formation at high redshifts. [ABSTRACT FROM AUTHOR]