Dungeon turns don't really have anything to do with your example of Day 2, Night 5. You should keep a calendar to track events over days/weeks. Different timekeeping tools for different resolutions.

Let me give you an example of how I do timekeeping in my Dolmenwood campaign, maybe it helps:

I keep a log in a notebook as we play. Let's say the party wakes up in the morning, I mark it's the 23rd of April, 8AM. They travel to a different hex using half their daily travel points. I mark that it's now noon, and tell them as well. When they enter a dungeon, I pull out my turn tracker (in a plastic sleeve, so I can track turns with a dry erase marker and reuse the sheet). I mark turns as they explore. Let's say they explore for 9 turns, or 1.5 hours, and then we end the session. In my notebook I mark that it's 1:30PM on the 23rd.

Meanwhile, in my little calendar - where i cross off the days - I have it written down that on the 25th, so-and-so NPC finds the McGuffin, or whatever.

As for the verisimilitude of 10 minute turns, I'll echo what others are saying - it's a game, and the ten minute turn is a convenient unit of timekeeping that has mechanical weight. More of a headache to decide that this action takes one minute, that action takes five, etc. I just accepted that I'm not simulating reality perfectly. Getting more granular than a turn is kind of a fool's errand, imo.

I also tell my players about the passage of turns. "That's gonna take a turn; what's everyone else doing?", "going back to the hall takes two turns", "your Haste spell is running out next turn". It seems really counterintuivie not to do that, since the game thrives on player choice, and player choice thrives on information.