We report a bright source of polarization-entangled photon pairs using spontaneous parametric down-conversion (SPDC) in a 10 mm long type-II PPKTP crystal pumped by a broadband multi-mode diode laser with the coherence length of 330 μm. Ordinarily, the huge mismatch between the pump coherence length and the PPKTP length would degrade the polarization entanglement completely. By employing the universal Bell-state synthesizer scheme, we remove the spectral/temporal distinguishability of the biphoton amplitudes entirely to recover high-visibility and high-fidelity two-photon polarization entanglement. The pair detection rates are 7,000 pairs/mW via single-mode fibers (with 99.2% fidelity) and 90,900 pairs/mW via multi-mode fibers (with 96.8% fidelity). We also analyze the scheme theoretically to show the effect of broadband multi-mode pumping on the phase matching condition of the type-II PPKTP.