This paper presents a novel approach for linear to circular polarization (LP-CP) conversion in transmitarray antennas. The proposed conversion mechanism differs significantly from previous published realizations. The concept utilizes a transmission line modeling-based excitation technique, in which a centralized via excitation is split into two striplines carefully designed to balance excitations and guarantee a phase quadrature. The striplines are embedded at the center of the patch antenna allowing a compact footprint and a simple design structure. The applied true-time delay (TTD) technique assures the radiation of a wideband and low axial ratio circularly polarized (CP) field. To optimize bandwidth and transmission efficiency, a stacked patch configuration is also employed, allowing for simultaneous high polarization conversion and transmission efficiency. The unit-cell design methodology is detailed, and two transmitarray designs are realized in both X and Ka bands. Experimental results from a fabricated X-band $20 \times 20$ cell array prototype demonstrate a peak aperture efficiency of 30%, accompanied by a simultaneous 16% -1 dB gain and 1 dB axial ratio bandwidths. Furthermore, the measured wideband and highly efficient Ka-band transmitarray with a $70 \times 70$ cell array confirmed a remarkable gain of 39.8 dB and 55% aperture efficiency at 29 GHz, surpassing previous LP-CP transmitarray antennas, while maintaining axial ratio values below 1 dB in a bandwidth larger than 26%.