SU($3$) Polyakov linear-sigma model (PLSM) is systematically implemented to characterize the quark-hadron phase structure and to determine various thermodynamic quantities and magnetization of the QCD matter. In mean-field approximation, the dependence of the chiral order-parameter on finite magnetic field is also calculated. In a wide range of temperatures and magnetic field strengths, various thermodynamic quantities including trace anomaly, speed of sound squared, entropy density, specific heat are presented and some magnetic properties are described, as well. Wherever available these results are confronted to recent lattice QCD calculations. The temperature dependence of these quantities confirms our previous result that the transition temperature is reduced with the increase in the magnetic field strength, i.e. QCD matter is to be characterized by an inverse magnetic catalysis. Furthermore, the temperature dependence of the magnetization shows that the conclusion that the QCD matter has paramagnetic properties slightly below and far above the pseudo-critical temperature, is confirmed, as well. The excellent agreement with recent lattice calculations proves that our QCD-like approach (PLSM) seems to possess the correct degrees-of-freedom in both hadronic and partonic phases and describes well the dynamics deriving confined hadrons to deconfined quark-gluon plasma.