Dickens later became friends with a Jewish family, and it was the wife of the family, in particular, who explained to Dickens how extremely offensive Fagin's portrayal was to her specifically, and British Jews more generally. He did edit out the many uses of "the Jew" in later editions (and it's probably the later editions you read in school, so you may have thought the Jewishness of Fagin was not really overplayed.)
But Dickens tried to make amends beyond editing the offending novel, by creating another Jewish character to counteract Fagin, and specifically to relate the lecture he must have gotten from his Jewish acquaintances.
The character was Mr. Riah from Our Mutual Friend (my favorite Dickens novel), a man who falls in with a non-Jewish moneylender. Mr. Riah had some financial troubles of his own, and became indebted to Fledgeby, who is looking to marry well himself (and gets thwarted and yadda yadda yadda). Fledgeby is a stereotypical dishonest moneylender, and uses Riah as the public face of his dirty work. If there's a bill to be called in, Riah is the one who has to do it face-to-face.
Riah never lies, and always says he is acting on the orders of the principal (though never telling anybody who his principal is, which is how Fledgeby maintains his social position).
But everybody assumes Riah is lying and is acting on his own behalf.
When this lie-by-assumption costs Riah a friend, he finally breaks and determines he must get free of Fledgeby. Once he settles with Fledgeby, he finds the friend, and explains the situation to her:
The little creature folded her arms about the old man's neck with great earnestness, and kissed him. 'I humbly beg your forgiveness, godmother. I am truly sorry. I ought to have had more faith in you. But what could I suppose when you said nothing for yourself, you know? I don't mean to offer that as a justification, but what could I suppose, when you were a silent party to all he said? It did look bad; now didn't it?''It looked so bad, Jenny,' responded the old man, with gravity, 'that I will straightway tell you what an impression it wrought upon me. I was hateful in mine own eyes. I was hateful to myself, in being so hateful to the debtor and to you. But more than that, and worse than that, and to pass out far and broad beyond myself—I reflected that evening, sitting alone in my garden on the housetop, that I was doing dishonour to my ancient faith and race. I reflected—clearly reflected for the first time—that in bending my neck to the yoke I was willing to wear, I bent the unwilling necks of the whole Jewish people. For it is not, in Christian countries, with the Jews as with other peoples. Men say, 'This is a bad Greek, but there are good Greeks. This is a bad Turk, but there are good Turks.' Not so with the Jews. Men find the bad among us easily enough—among what peoples are the bad not easily found?—but they take the worst of us as samples of the best; they take the lowest of us as presentations of the highest; and they say "All Jews are alike." If, doing what I was content to do here, because I was grateful for the past and have small need of money now, I had been a Christian, I could have done it, compromising no one but my individual self. But doing it as a Jew, I could not choose but compromise the Jews of all conditions and all countries. It is a little hard upon us, but it is the truth. I would that all our people remembered it! Though I have little right to say so, seeing that it came home so late to me.'The dolls' dressmaker sat holding the old man by the hand, and looking thoughtfully in his face.'Thus I reflected, I say, sitting that evening in my garden on the housetop. And passing the painful scene of that day in review before me many times, I always saw that the poor gentleman believed the story readily, because I was one of the Jews—that you believed the story readily, my child, because I was one of the Jews—that the story itself first came into the invention of the originator thereof, because I was one of the Jews. This was the result of my having had you three before me, face to face, and seeing the thing visibly presented as upon a theatre. Wherefore I perceived that the obligation was upon me to leave this service. But Jenny, my dear,' said Riah, breaking off, 'I promised that you should pursue your questions, and I obstruct them.'
Obviously, they are reconciled after this little speech.
However, Riah's speech is such that one gets the feeling this is what Dickens himself was told, in the offense of Fagin the Jew. Sure, Fagin was based on a real person who happened to be a Jew, but Ikey Solomon's criminality as a fence didn't have much to do with his Jewishness, and Oliver Twist overly emphasized it. As Riah said, instead of Jewish readers seeing Fagin as a bad Jew, they saw Fagin as intended as an indictment against all Jews. Dickens did eventually see it that way and tried to make amends.
However, Riah is a bit too much for amends. Earlier in the day, Jenny happened upon a scene wherein somebody found out about the truth about Fledgeby and caned him mercilessly. She came in right after, and found Fledgeby in pain and "helped" him (I will not get into details, but that's when she also realizes the truth, about Fledgeby, at any rate. The truth about Riah had to come from him directly.) She explains it to Riah and he decides that, even after breaking off his situation with Fledgeby, he really should go help him.
'I mean, godmother,' replied Miss Wren, sitting down beside the Jew, 'that the fox has caught a famous flogging, and that if his skin and bones are not tingling, aching, and smarting at this present instant, no fox did ever tingle, ache, and smart.' Therewith Miss Jenny related what had come to pass in the Albany, omitting the few grains of pepper.
.....This expression of regret being but partially intelligible to Mr Riah, the old man reverted to the injuries Fledgeby had received, and hinted at the necessity of his at once going to tend that beaten cur.'Godmother, godmother, godmother!' cried Miss Wren irritably, 'I really lose all patience with you. One would think you believed in the Good Samaritan. How can you be so inconsistent?''Jenny dear,' began the old man gently, 'it is the custom of our people to help—''Oh! Bother your people!' interposed Miss Wren, with a toss of her head. 'If your people don't know better than to go and help Little Eyes, it's a pity they ever got out of Egypt. Over and above that,' she added, 'he wouldn't take your help if you offered it. Too much ashamed. Wants to keep it close and quiet, and to keep you out of the way.'
It turns out that Jenny is correct on this score. A note arrives right after this to tell Riah to get the hell out and never see him again.
One does get a bit out of patience with these over-the-top saints. Come on, Dickens.