Ultimately the text in this passage has undergone considerable shifting towards the present tense. The transition began when Oliver Cowdery accidentally changed the past-tense crucified to crucify when he copied the text from 𝓞 to 𝓟. Then Joseph Smith, in his editing for the 1837 edition, changed the conjoined past-tense verb form turned to turn (thus producing “and turn their hearts aside”). Later, in the next verse, while copying from 𝓞 to 𝓟, Oliver also accidentally dropped the present perfective have from “because they have turned their hearts aside”. Thus in his editing for the 1837 edition, Joseph was again confronted with the past-tense form turned, which he once more edited to the present-tense turn. Instead, the original have, if it had been restored, would have brought back the original parallelism in verse 14 between the two perfective verb phrases have turned and have despised. But Joseph did not examine 𝓞 in his editing for the 1837 edition.
The original past-tense forms in verse 13 are acceptable because by the time the people of Jerusalem “shall be scourged by all people”, they will have already “crucified the God of Israel and turned their hearts aside”. The main reason Joseph Smith edited the two occurrences of the pasttense form turned was because they conflicted with the preceding present-tense crucify, which was an accident in the first place.
Summary: Restore in 1 Nephi 19:13–14 the past-tense crucified and turned as well as the present perfect have turned, the readings of the original manuscript.