Penal substitution requires at least two things (1) that Jesus’ death is just, or deserved, that is, to satisfy the wrath of God, to satisfy the retributive demands of God’s justice, and (2) that Jesus dies in our place, as our substitute, taking the punishment upon himself so we won't have to suffer it. Isaiah 53 fails both these criteria.
First, let’s pay close attention the logic of Isaiah 53:5, that “by his scourging we are healed.” The logic of penal substitution is “Jesus suffered the punishment for our sins in our place so that we would not have to suffer that punishment.” In the language of Isaiah 53, this would be translated “Jesus was wounded so that we would avoid being wounded.” But this is not what Isaiah 53 says. The text says, “By his scourging we are healed,” NOT, “By his scourging we avoid being scourged.” The logic of Isaiah 53 actually runs, “He was wounded so that we could be healed from our wounds.” The suffering servant is wounded so that those who are wounded (by sin) can be healed. Jesus is wounded for our transgressions so that our wounds can be healed by the power of his resurrection. Again, Jesus does not die so that we will not have to die, but Jesus dies so that the dead can rise in Him. See Ephesians 2:1-10. Our problem is we are dead, the solution is resurrection in Christ.
Second, many penal substitution advocates will cite that Jesus was “smitten of God (v.4)” and “the Lord was pleased to crush him (v.7).” But recall that God’s act of “crushing” people can refer to two different things, both (a) the just punishment of the guilty, or (b) the unjust persecution of the innocent. See Psalm 44, in which the Psalmist says, “Our heart has not turned back, and our steps have not deviated from Your way, yet You have crushed us in a place of jackals (referring to foreign nations) and covered us with the shadow of death (v.18-19).” The Psalmist says that God is crushing him, rejecting him, dishonoring him (v.9), yet he is innocent, so he is clearly referring to unjust persecution at the hands of the wicked, not just punishment at the hands of the righteous, when he says so. Therefore, references to God “striking” and “crushing” the suffering servant fail to show that the suffering servant is justly suffering or “satisfying the wrath” of God. Isaiah 53 is about an innocent person who is crushed by God as he willingly suffers the unjust persecution of wicked humans.
Third, it is clear from the immediate context, the sacrificial context, and from the commentary on the passage in 1 Peter 2 that the Suffering Servant is suffering unjustly, not justly. Concerning the immediate context of Isaiah 53, the prophet makes it clear that the Suffering Servant is “despised and forsaken by men” (v.3), and “they made [the Servant’s] grave with the wicked . . . although he had done no violence, and there was no deceit in his mouth.” These verses make it clear that the Servant was unjustly killed. Concerning the sacrificial context of Isaiah 53, verse 10 says that the Servant was “made a guilt offering.” Recall that a guilt offering is a ritual in which an unblemished animal is killed by a guilty sinner (Lev 5:6). The Servant, also, is a perfectly innocent life that is killed by guilty sinners. Concerning the commentary given to us in 1 Peter 2, notice that Isaiah 53 is the go-to passage for encouraging people who are unjustly suffering. Peter says:
“Servants, be submissive to your masters with all respect, not only to those who are good and gentle, but also to those who are unreasonable. For this finds grace, if for the sake of conscience toward God a person bears up under sorrows when suffering unjustly. For what credit is there if, when you sin and are harshly treated, you endure it with patience? But if when you do what is right and suffer for it you patiently endure it, this finds grace with God.
For you have been called for this purpose, since Christ also suffered for you, leaving you an example for you to follow in His steps, Who committed no sin, nor was any deceit found in his mouth; and while being reviled, He did not revile in return; while suffering, He uttered no threats, but kept entrusting Himself to Him who judges righteously; and He Himself bore our sins in His body on the cross, so that we might die to sin and live to righteousness; for by His wounds you were healed.”
Lastly, notice how Acts 8 translates the passage of Isaiah 53:
“Like a sheep he was led to the slaughter
and like a lamb before its shearer is silent,
so he opens not his mouth.
33 In his humiliation justice was denied him.
Who can describe his generation?
For his life is taken away from the earth.”
This explicitly states that justice was denied Christ in his death. His death was unjust.