Regularized reduced order models (Reg-ROMs) are stabilization strategies that leverage spatial filtering to alleviate the spurious numerical oscillations generally displayed by the classical Galerkin ROM (G-ROM) in under-resolved numerical simulations of turbulent flows. % In this paper, we propose a new Reg-ROM, the time-relaxation ROM (TR-ROM), which filters the marginally resolved scales. % We compare the new TR-ROM with the two other Reg-ROMs in current use, i.e., the Leray ROM (L-ROM) and the evolve-filter-relax ROM (EFR-ROM), in the numerical simulation of the turbulent channel flow at $Re_{\tau} = 180$ and $Re_{\tau} = 395$ in both the reproduction and the predictive regimes. % For each Reg-ROM, we investigate two different filters: (i) the differential filter (DF), and (ii) a new higher-order algebraic filter (HOAF). % In our numerical investigation, we monitor the Reg-ROM performance with respect to the ROM dimension, $N$, and the filter order. We also perform sensitivity studies of the three Reg-ROMs with respect to the time interval, relaxation parameter, and filter radius. The numerical results yield the following conclusions: % (i) In terms of the Reynolds normal and shear stresses, all three Reg-ROMs are significantly more accurate than the G-ROM. (ii) In addition, all three Reg-ROMs are more accurate than the ROM projection, which represents the best theoretical approximation of the training data in the given ROM space. (iii) With the optimal parameter values, the new TR-ROM yields more accurate results than the L-ROM and the EFR-ROM in all tests. (iv) For most $N$ values, DF yields the most accurate results for all three Reg-ROMs. (v) The optimal parameters trained in the reproduction regime are also optimal for the predictive regime for most $N$ values, demonstrating the Reg-ROM predictive capabilities. (vi) All three Reg-ROMs are sensitive to the filter radius and the filter order, and the EFR-ROM and the TR-ROM are sensitive to the relaxation parameter. (vii) The optimal range for the filter radius and the effect of relaxation parameter are similar for the two $\rm Re_\tau$ values.