15 year old Kirk was there. He wrote down the Kodos speech from memory for Karidian to recite for a speech analysis.
KIRK: You're an actor now. What were you twenty years ago?
KARIDIAN: Younger, Captain. Much younger.
KIRK: So was I. But I remember. Let's see if you do. Read this into that communicator on the wall. It will be recorded and compared to a piece of Kodos' voice film we have in our files. The test is virtually infallible. It will tell us whether you're Karidian, or Kodos the Executioner. (switches on comm.) Ready for voice test. Disguising your voice will make no difference.
KARIDIAN: (reading) The revolution is successful, but survival depends on drastic measures. Your continued existence represents a threat to the well-being of society. (stops looking at the paper) Your lives means slow death to the more valued members of the colony. Therefore I have no alternative but to sentence you to death. Your execution is so ordered. Signed, Kodos, governor of Tarsus Four.
KIRK: I remember the words. I wrote them down. You said them like you knew them. You hardly glanced at the paper.
In "The Deadly Years" the computer said that Kirk's physiological age was about 60 to 72 years:
COMPUTER: Working. Subject's physical age based on physiological profile, between sixty and seventy two. Aging rapidly.
KIRK: No, I'm thirty four. I'm thirty four years old.
Since "The Conscience of the King" was a first season episode it was probably more or less a year earlier than "The deadly Years". So Kirk was probably about thirty three years old in "The Conscience of the King", which was 20 years after the Tarsus IV incident, thus making Kirk about thirteen during the Tarsus IV incident.
"Shore Leave" was also a first season episode, and Kirk should have been about thirty three in "Shore Leave".
KIRK: What's the matter, Bones, you getting a persecution complex?
MCCOY: Well, yeah, I'm beginning to feel a little bit picked on, if that's what you mean.
KIRK: I know the feeling very well. I had it at the Academy. An upper classman there. One practical joke after another, and always on me. My own personal devil. A guy by the name of Finnegan.
MCCOY: And you being the very serious young
KIRK: Serious? I'll make a confession, Bones. I was absolutely grim, which delighted Finnegan no end. He's the kind of guy to put a bowl of cold soup in your bed or a bucket of water propped on a half-open door. You never knew where he'd strike next. More tracks. Looks like your rabbit came from over there.
Kirk also meets a former girlfriend Ruth:
KIRK: (into communicator) McCoy, do you read me? Ruth. Ruth, how can it be you? How could you possibly be here? You haven't aged. It's been fifteen years.
Later:
SPOCK: There is one slight possibility, very slim, but nevertheless. Captain, what were your thoughts just before you encountered the people you described?
KIRK: I was, I was thinking about the Academy. My days
When Kirk meets Finnegan for the second time:
FINNEGAN: Get up. Get up. Get up. Always fight fair, don't you? True officer and gentleman, you. You stupid underclassman. I've got the edge. I'm still twenty years old. Look at you. You're an old man.
Since Finnegan was an upperclassman, he should have graduated before Kirk did. Finnegan is twenty years old, the way Kirk remembers from the first time they met, or the last time, or sometime in between.
FINNEGAN: I never answer questions from plebes, Jimmy boy.
KIRK: I'm not a plebe. This is today, fifteen years later. What are you doing here
A plebe is a term for a first year cadet. So Kirk was probably a first year cadet fifteen years before he was about thirty three, and thus when aged about eighteen. The age range to enter US service academies is seventeen to twenty two, and the most common age is eighteen. So if Kirk was about eighteen fifteen years before "Shore Leave", he should have been about thirteen twenty years before "The Conscience of the King" which should happen about the time of "Shore Leave".
Memory Alpha also says that Kirk was thirteen at Tarsus IV.
https://memory-alpha.fandom.com/wiki/James_T._Kirk
And for whatever it might be worth, the official but non canon and possibly incorrect
Star Trek Chronology: The History of the Future, 1996, says Kirk was born in 2233 and the Tarsus IV Massacre happened in 2246, the year that Kirk turned thirteen.
So the evidence indicates that Kirk was about thirteen, not fifteen, during the events on Tarsus IV.
