Matthew 12:20: A battered reed he will not break off; and a smoldering wick he will not put out, until he leads justice to victory.

This verse is quoted from a prophecy in Isaiah 42:1-3, and according to Matthew 12, clearly illustrates the approach and attitude of Jesus in his earthly ministry.

Matthew 12:14-20: But the Pharisees went out, and counseled together against him, as to how they might destroy him. But Jesus, aware of this, withdrew from there. And many followed him, and he healed them all, and wanted them not to make him known, in order that what was spoken through Isaiah the prophet, might be fulfilled, saying, Behold, my servant whom I have chosen; my beloved in whom my soul is well-pleased; I will put my spirit upon him, and he shall proclaim justice to the Gentiles (the nations). He will not quarrel, nor cry out, nor will anyone hear his voice in the streets.

A battered reed he will not break off; and a smoldering wick he will not put out, until he leads justice to victory.

Jesus was specially chosen and beloved of God. He was led and directed by the spirit of God, with the end result of bringing about justice to the people. 

Matthew 12:19: He will not quarrel, nor cry out, nor will anyone hear his voice in the streets.

He was led by the spirit—the power and influence of God—to be calm and confident as he preached truth, not like the presentation of the spiritual leadership of Israel whowas argumentative and full of false accusation.  

Matthew 7:28-29: When Jesus had concluded this discourse, the crowds were filled with amazement at his teaching, for he had been teaching them as one who had authority, and not as their scribes taught.

Luke 6:7: The scribes and the Pharisees were watching him closely to see if he healed on the Sabbath, so that they might find reason to accuse him.

More specifically on your question:

He was very understanding of the difficulties and sincere limitations of the people, and he had compassion on the emotional and spiritual wounds and the limited light of truth they were able to shine forth. He was sensitive to the sincere vulnerabilities and challenges of the people.  

Mathew 9:36: Whenever crowds came to him, he had compassion for them because they were so deeply distraught, malaised, and heart-broken. They seemed to him like lost sheep—like sheep without a shepherd.

Again, this in is contrast to the spiritual leadership of that day. Instead of trying to help people, they wanted power and set unreasonable and unscriptural standards that they themselves did not keep. 

Matthew 23:1-4: The scribes and the Pharisees have seated themselves in the chair of Moses; therefore all that they tell you, do and observe, but do not do according to their deeds, for they say things, and do not do them. And they tie up heavy loads, and lay them on men’s shoulders; but they themselves are unwilling to move then with so much as a finger.

Jesus saw beyond the background and limitations of a person, and encouraged their spiritual seeking, whereas the leadership judged by the appearance of things. 

Luke 5:27-30: And after that he went out, and noticed a tax-gatherer named Levi, sitting in the tax office, and he said to him, Follow me. And he left everything behind, and rose and began to follow him. And Levi gave a big reception for him in his house; and there was a great crowd of tax-gatherers and other people who were reclining at the table with them. And the Pharisees and their scribes began grumbling at his disciples, saying, Why do you eat and drink with the tax-gatherers and sinners? And Jesus answered and said to them, It is not those who are well who need a physician, but those who are sick.

A battered reed he will not break off; and a smoldering wick he will not put out, until he leads justice to victory.

The end result for many of the “lost” would be the victory of righteousness and what is right.              

A future fulfillment of these verses is also taught.

Isaiah 42:4 adds another verse that is not quoted in the Matthew account, which illustrates that this supportive and patient attitude would also be exercised when the kingdom of God is fully established on the earth. Isaiah 42:4: He will not be disheartened or crushed, until he has established justice in the earth; and the coastlands will wait expectantly for his law.

This verse alludes to a conversion of the whole world. Isaiah 11:9: They will not hurt or destroy in all My holy mountain, for the earth will be full of the knowledge of the LORD as the waters cover the sea. 

This kingdom, the rulership of Christ and his faithful followers, his “bride,” will resurrect all those born in Adam who were not faithful Christians. 

John 5:28-29: Do not marvel at this; for an hour is coming, in which all who are in the tombs shall hear his voice, and shall come forth; those who did the good deeds to a resurrection of life, those who committed the evil deeds to a resurrection of judgment. 

The resurrection of judgment will lead to the full opportunity for all the effects of sin–of being ‘battered’ or abused and wounded in heart and mind, to SEE the truth and to accept Christ and to walk the ‘highway of holiness.’ This highway of holiness will be a symbolic journey of learning and a process of literal and spiritual healing, which will lead to development of a character of holiness. Isaiah 35:5-6,8: Then the eyes of the blind shall be opened, and the ears of the deaf shall be unstopped. Then shall the lame leap as an hart, and the tongue of the dumb sing…and an highway shall be there, and a way, and it shall be called the way of holiness; the unclean shall not pass over it, but it shall be for those; the wayfaring men, though fools, shall not err therein.”

What a blessing that the compassion of Christ described in Matthew 12:20 is lavished upon the followers of Christ now, and on the whole rest of our human family in the future!