Half H-Bridge Driver
Your circuit won't work because you have failed to account for the fact that MOSFETs are controlled by the voltage DIFFERENCE between their gate and source pins. The voltage you are applying to the gate is relative to ground, so if the source pin is also connected to ground then you have no problems since the voltage you are trying to apply to the gate is relative to ground, and the voltage the MOSFET cares about (the gate-source voltage) has been forced to be relative to ground (like with Q2 and Q3).

But this is not the case with Q1 and Q4. Suppose Q1 or Q4 did actually turn on properly and current flows through the load. A voltage drop develops across the load with one end of the load being connected to ground (via Q2 or Q3). So what happens to the voltage of the source pin of Q1 or Q4 relative to ground? How does this affect the difference of the voltage between the gate-source? And what happens if you are applying a gate voltage that is relative to ground, but the source pin voltage relative to ground is no longer at ground?

Try and think about it carefully and see if you understand why Q1 or Q4 can never actually turn on completely or properly. If you are stuck, see here:
Why choose a PMOS over an NMOS or vice versa?

Think something like a source follower.

Or think about an electromechanical relay. It has two coil pins for control and two contact pins for the primary power flow. You would never ever apply a signal to one of the coil pins while ignoring the other coil pin and expect it to work right?

A transistor is the same, except one of the coil pins and one of the primary contact pins is fused together (the source pin on a MOSFET is a bit like a fused version of a relay coil pin and contact pin). That means you cannot apply a voltage to the gate and expect it to work without taking into account the relationship with the source pin. That's pretty much what you are doing here.

Unlike a relay, since a control and primary current flow pin are fused in a transistor, it means that any load connected to the source pin can influence the requirements of the control signal so your gate drive requirements can get trickier.

So you either need a gate-drive that supports a bootstrap capacitor and and diode use it with a bootstrap capacitor, or substitute the bootstrap capacitor/diode with an isolated regulator that floats (more expensive but is required to allow 100% duty cycle since bootstrap capacitors require the Q1 and Q4 to periodically turn off and Q2 and Q3 to periodically turn on in order to refresh the charge on the boostrap capacitor (this provides a connection that connects the bootstrap capacitor between GND and the power supply to recharge it). This last paragraph probably won't make too much sense to you until you go and find a high-side gate driver IC and look at the schematic for. So go do that.