Here is a link to an article I wrote that directly addresses your question: https://www.physicsforums.com/insights/reversible-vs-irreversible-gas-compressionexpansion-work/
It starts out by analyzing reversible vs irreversible work in a purely mechanical system involving a spring and a damper (dashpot) in parallel. This is like the spring and shock absorber system in your car. It then discusses the complete analogy between such a system and that of a gas experiencing reversible or irreversible compression or expansion. I hope it will give you a good fundamental understanding of the physical mechanisms that are involved.
ADDENDUM: Adiabatic irreversible expansion at constant pressure
In this case, you suddenly drop the external pressure from P1 to P2, and then hold the external pressure at P2 until the system equilibrates. This is what we call constant-pressure irreversible expansion.
$$nC_v(T_2-T_1)=-P_{ext}(V_2-V_1)=-P_2(V_2-V_1)=-P_2\left(\frac{nRT_2}{P_2}-\frac{nRT_1}{P_1}\right)$$
I replace $P_{ext}$ with $P_2$ because that is the final pressure that the gas equilibrates to. Solve the equation for $T_2/T_1$ as a function of $P_2/P_1$. Then determine $V_2/V_1$.
For reversible adiabatic expansion, we drop the external pressure gradually such that $$nC_vdT=-PdV=-\frac{nRT}{V}$$ In this case, the external pressure is always equal to that determined for the gas by the ideal gas law.
ADDENDUM #2
Solution for adiabatic reversible expansion or compression
$$P_{ext}(V)=P(V)=P_1\left(\frac{V_1}{V}\right)^{\gamma}$$
Solution for isothermal reversible expansion or compression
$$P_{ext}(V)=P(V)=P_1\left(\frac{V_1}{V}\right)$$
Solution for adiabatic reversible expansion or compression at constant external pressure from initial volume $V_1$ to final equilibrium volume $V_2$
$$P_{ext}(V)=\frac{P_1}{1+\gamma\left(\frac{V_2}{V_1}-1\right)}$$This equation applies for all $V_1<V<V_2$ in expansion or $V_2<V<V_1$ in compression
Solution for reversible expansion or compression of a gas held in contact with a constant temperature reservoir at the initial gas temperature $T_1$ and held at a constant external pressure from initial volume $V_1$ to final equilibrium volume $V_2$ $$P_{ext}(V)=P_1\frac{V_1}{V_2}$$
This equation applies for all $V_1<V<V_2$ in expansion or $V_2<V<V_1$ in compression