Among evangelical Protestants, especially, but also in various other quarters of Christianity, it is common to believe that "Jesus died for me" in the sense that Jesus' death made satisfaction for (in Catholicism) or specifically made satisfaction by paying the penalty for (in Protestantism) my individual sins.
There is a great deal of wording in the Bible supporting the idea that Jesus died for the sins of humans collectively. For example, this passage from Isaiah is commonly interpreted by Christians as referring to Jesus' death for us on the Cross:
Surely he has borne our infirmities
and carried our diseases;
yet we accounted him stricken,
struck down by God, and afflicted.
But he was wounded for our transgressions,
crushed for our iniquities;
upon him was the punishment that made us whole,
and by his bruises we are healed.
All we like sheep have gone astray;
we have all turned to our own way,
and the Lord has laid on him
the iniquity of us all.
The language here is not individual, but collective: "our diseases," "our transgressions," "has made us whole," "all we like sheep," "the iniquity of us all."
For while we were still weak, at the right time Christ died for the ungodly. Indeed, rarely will anyone die for a righteous person—though perhaps for a good person someone might actually dare to die. But God proves his love for us in that while we still were sinners Christ died for us. (Romans 5:6–8)
Here dying for an individual righteous person is mentioned as a parallel example, but the language specifically about Jesus' death is still collective: "While we were still weak," "died for the ungodly [the Greek is plural]," "while we were still sinners Christ died for us."
I am specifically not asking for the answer, "Jesus died for everyone, and that means he died for you, too." That answer says, essentially, "You are part of the collective, so Jesus' death covers you as well."
Rather, I am asking for the biblical basis for the belief that Jesus' death made satisfaction for individual sins, not just for the collective sins of humanity under which individuals are also covered. To use a legal analogy: that Jesus did not engage in a class action lawsuit, but litigated—and litigates—each individual's sins separately. Pointing out that particular individuals belong to the class of "sinners" does not answer this question.
In layman's terms, what is the biblical basis for the belief that Jesus suffered and died for each one of my individual sins, and not just for the sins of all of humanity, which covers my particular sins also?
Note 1: Though the question "What is the Biblical basis for thinking that Jesus died for me specifically?" is closely related to this one, as the OP says in a comment on the accepted answer, that question is more about whether Jesus was consciously thinking of, and dying, for, individuals while he was on the cross. My question is not concerned with Jesus conscious awareness (or lack thereof) of dying for every individual's sins on the cross, but with the biblical basis of the belief that Jesus' death made satisfaction for specific sins committed by specific individuals, and not just general satisfaction for the collective sins of humanity.
Note 2: I am asking this question from the perspective of those denominations, mainly Catholic and Protestant, that subscribe to one or another form of satisfaction theory, and that may include the belief that this means Christ made satisfaction for individual, not just collective, sins. I am presuming that the question applies primarily to evangelical Protestantism, but the answer should not be significantly different if answered from a Catholic satisfaction model that includes the idea that Jesus died to make satisfaction for individual sins. If even today the Catholic Church does not hold that Jesus made satisfaction for individual sins, but only for collective ones, please provide me with the relevant references to Catholic doctrine on that subject, and I'll edit it out of the question.