We showed in our previous studies that just 3% cross-links (CLs), at special points along the contour of the bacterial DNA, help the DNA-polymer to get organized at micron length scales [T. Agarwal et al., J. Phys.: Condens. Matter 30, 034003 (2018) and T. Agarwal et al., EPL (Europhys. Lett.) 121, 18004 (2018)]. In this work, we investigate how does the release of topological constraints help in the "organization" of the DNA-polymer. Furthermore, we show that the chain compaction induced by the crowded environment in the bacterial cytoplasm contributes to the organization of the DNA-polymer. We model the DNA chain as a flexible bead-spring ring polymer, where each bead represents 1000 base pairs. The specific positions of the CLs have been taken from the experimental contact maps of the bacteria Caulobacter crescentus and Escherichia coli. We introduce different extents of ease of release of topological constraints in our model by systematically changing the diameter of the monomer bead. It varies from the value where chain crossing can occur freely to the value where chain crossing is disallowed. We also study the role of compaction of the chain due to molecular crowders by introducing an "effective" weak Lennard-Jones attraction between the monomers. Using Monte Carlo simulations, we show that the release of topological constraints and the crowding environment play a crucial role to obtain a unique organization of the polymer.