Well, it can also be used, but doesnt "go on with" tell the same thing? ı am really asking-sincerely! here he/she is asking to go on the conversation in Turkish, isn't he/she?
the verb "geçmek" can be used as so many meanings -not just "switch to"
text shows: they are having a chat, but as they can't find anything to talk about any more, he is asking to chat in Turkish -as they cant speak in the other language they have already been talking. and in this situation, "go on with" can also gives the same idea/meaning or whatever.
well, that's all what I understood from the text
and btw, thank you too merdogan for your comment..