H-bridge design for Peltier
Peltier modules don't like being driven by PWM; see this question.  Peltier coolers suffer from \$I^2R\$ losses; if you drive them with a continuous voltage then those losses are minimized; drive them with PWM and those losses are maximized.

This means that you need to drive your module with a class D amplifier that has output filtering.  The "how to" for that exceeds the available space here, but you might want to start with this app note about class D audio amplifiers and this app note about driving a Peltier module with a class D amplifier.  At a minimum you need to drive all four gates in your H bridge with proper gate drivers, you need at least series-L filters on the H bridge output (with adequate inductors), and you need to mind your p's and q's with regards to board layout.