Your English seems flawless to me. (But in this context, instead of "a dialog turning into a fight", I would probably say "a conversation turning into an loud argument".
Викисловарь offers this figurative meaning of "базар":
2. перен. шум, крики, громкая брань
This makes sense to me, because when I was an American child living in Turkey, we would buy most of our bread, cheese, vegetables, and fruits at the Turkish "pazar", which was always a noisy, chaotic place.
(But usually we bought fresh meat, eggs, and milk through the American military commissary, where the hygiene standards were much more strict!)
So, as a very loose "video translation" of this verse:
А ларечек тот типа барчика
Слово за слово до базарчика
Из девяточки русский рэпачок
Нужен ножечек, если не качек
...позвольте меня представить этот ролик, из 1980-го амер. к/ф "Airplane!":
"The mood in the place was downright ugly...
"You wouldn't walk in there unless you knew how to use your fists...
"You could count on a fight breaking out almost every night."