The most recently created extra turn is taken first. Since the copy resolves before the original, your opponent gets to take his extra turn first. Then you get your extra turn, then the normal turn order resumes. So yes, your guess is correct as long as it is his turn. If he casts TM during your turn (like via miracle), you continue with your current turn, then take an extra turn, then he takes an extra turn, then he gets his normal turn. then he takes an extra turn, then you take your extra turn, then he gets his normal turn.
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"Simple questions" usually need rather complex answers, while complex questions often come down to no more than a simple "yes" or "no".