If someone wants to turn a conversation into political way, we call it "politicize"
If someone wants to turn a conversation into drama, we call it "dramatize"
What do we call it when someone wants to turn a conversation into logical conversation?
How about logicalize? From M-W:
logicize: to make logical : convert to logical form, e.g. logicized the argument
While logicalize is obviously similar to @Noah's answer, "to logicalize a conversation" seems a natural way to describe "the turning of a conversation into a logical conversation".
I think "logicize" may be the correct word.
This dictionary.com entry seems to back that idea up (https://www.dictionary.com/browse/logicize)
The definition states "to make logical; give logical form to".
The definition could be talking about the application of logic abstractly or to something specific.