All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
There is nothing wrong in regard to the method of reciting the Quran that you mentioned, whether you meant by saying 'take turns' that each one of them recites two verses that were recited by the one before him or you meant that each one of them starts from where the previous one had stopped and then recites two verses that follow: all this is permissible and there is nothing wrong with it. The evidence of this is that the Prophet, sallallaahu ‘alayhi wa sallam, said, “Whenever people assemble in one of the houses of Allaah (mosques) and recite the Book of Allaah and learn and teach the Quran (among themselves), tranquility descends upon them, mercy covers them, the angels surround them, and Allaah makes a mention of them in the presence of those near Him…” [Muslim]
This has nothing to do with what Ibn Mas'ood admonished the Kharijites about, as they used to sit in circles and one of them would order them to praise Allaah with a certain number of praises.
There are four methods of reciting the Quran:
The first method: one person recites and the others listen to him: this method is recommended without any difference of opinion. Abdullaah ibn Mas'ood narrated that the Prophet told him, “Recite the Quran to me.” I [Ibn Mas’sood] said, “Would I recite the Quran to you while it was revealed to you? He said, “I like to hear it from someone else.” I recited Chapter An-Nisaa' [Quran 4] to him, and when I reached (what means) {So how [will it be] when We bring from every nation a witness and we bring you [O Muhammad] against these [people] as a witness?}, he said, “Stop.” And I saw tears in his eyes.” [Al-Bukhari and Muslim]
Ibn Taymiyyah said, “With regard to one person reciting while the rest are listening to him, then this is not disliked without any difference of opinion, rather this is recommended; this is what the companions used to do, such as Abu Moosa and others.”
The second method: a reciter recites some verses from the Quran and then stops, and then another one would repeat what the first one had recited in order to study the Quran: this method is recommended according to the agreement of the scholars because Jibreel (Gabriel) used to teach the Quran to the Prophet in Ramadaan, and each one of them would recite to the other. Ibn Hajar said in al-Fat-h, “The Prophet used to recite the Quran to Jibreel. This is contrary to the chapter title (in Saheeh Al-Bukhari), as the title suggests that Jibreel used to recite to the Prophet and here it is the Prophet who used to recite to Jibreel. It was mentioned in regard to the beginning of revelation that 'he would meet him every night of Ramadaan and recite to him the Quran'; this may be understood either way: that each of them used to recite to the other."
The third method: a reciter recites and then stops, and then another one continues from where the first one stopped; the majority of the scholars hold the view that this is good and it is not disliked. But the Hanbali School hold the view that this method is disliked. Ibn Muflih said in Al-Furoo', “Our scholars dislike the recitation in which one starts and then the other continues, and Harb said, "It is good, and our Shaykh (i.e. Shaykh Ibn Taymiyyah ) quoted this from most scholars.”
The most preponderant opinion about this method is that it is not disliked because Abu Hurayrah narrated that the Prophet said, “Whenever people assemble in one of the houses of Allaah (mosques) and recite the Book of Allaah and learn and teach the Quran (among themselves), then tranquility descends upon them, mercy covers them, the angels surround them, and Allaah makes a mention of them in the presence of those near Him.” [Muslim and Abu Daawood]
Since the Hanbali School agreed with the majority of the scholars that the second method is recommended, then there is no reason for this method to be disliked, because there is no difference between the reciter repeating what the first one had recited or continuing from where the first one had stopped. Shaykh Ibn Taymiyyah said, “The recitation whereby one recites and when he finishes another one continues is good according to most scholars.”
Besides, Imaam An-Nawawi said in at-Tibyaan, “Chapter in reciting the Quran in turns, which is that a group gather, and one of them would recite a tenth or a juz’ [part of the Quran], and so forth, and then he stops and then another one continues to recite from where the first one had stopped, and then another one recites and so forth; this is permissible and good. Imaam Maalik was asked about it and he said, 'It is acceptable.'”
The fourth method: the whole group recites in one voice. The Hanbali and Shaafi'i Schools of jurisprudence hold the view that it is recommended, and this is the second opinion of the Hanafi School. Shaykh Ibn Taymiyyah, said, “Reciting by taking turns is good according to most scholars, and among reciting by taking turns is reciting altogether with one voice, and the Maaliki School has two opinions about its dislikability.”
The second view is that reciting in a group together with one voice is dislikable as this involves abandoning listening and confusing each other; this is the reliable view of the Hanafi and Maaliki Schools, so this last method should be avoided, as all goodness lies in following the Salaf (righteous predecessors); and one should avoid boasting, showing off, the mixing of sounds, and abandoning listening.
Finally, what you have mentioned that the Companions used to return to their homes as soon as the prayer ends, then we are not aware of any evidence regarding this; rather, the Sunnah proves otherwise, that they used to sit with the Prophet in the mosque after Fajr until sunrise. Simaak ibn Harb said, “I said to Jaabir ibn Samurah, 'Were you used to sit with the Prophet ?' He replied, 'Yes, often. He used to not get up from his place where he had prayed the Fajr prayer until the sun would rise, and when the sun would rise, then he would get up, and the companions would recall the stories of the pre-Islamic era of ignorance and they would laugh, and the Prophet would smile.'” [Muslim]
An-Nawawi said, “This is evidence that it is desirable to mention Allaah after the Fajr prayer and to sit where one has prayed the Fajr prayer unless one has a sound reason to leave.” Al-Qaadhi said, “This is a Sunnah which the Salaf and the scholars used to do.”
Allaah knows best.