That works for motors which generate back-EMF from stored rotational energy, but not for a largely inductive load with stored inductive energy.To quickly turn off the solinoid after activating it, both low-side mosfets can be activated (without activating high side). Works well for dynamic braking on motor controllers.

One option to reconsider for in such application is to use a common Power Op-Amp IC's for your H-Bridge. They work well as dedicated H-bridge devices plus give you a lot of other possible control options with minimal parts counts and costs.My real load is a solenoid, not a motor. The measured voltages in the previous post were with a DMM as the load. The measured resistance of the solenoid as 4 ohms, which at face value should require 3 amps at 12V, however it consistently actuates with a relay sourced by a 12V 200mA supply.
I doubt they will work well in a digital bridge to switch a solenoid since they likely can't go fully to the supply voltages at their outputs and will have a relatively slow switching speeds, causing poor efficiency.One option to reconsider for in such application is to use a common Power Op-Amp IC's for your H-Bridge. They work well as dedicated H-bridge devices plus give you a lot of other possible control options with minimal parts counts and costs.
The tried and true single channel LM1875 Power Op amp IC can easily handle 4 amps and costs a bit over $3.
Or the higher capacity dual channel LM3886 can handle 7 amps and cost ~$7.