This is the 3055th most frequent Turkish word.
bayrağını
"Bayrağını" means "your flag" in English.
Here, 'bayrağını' means 'his/her flag', and it is used as an object in the action of 'waving'.
Ali bayrağını dalgalandırarak meydanda yürüyordu.
Ali was walking in the square waving his flag.
In this sentence, 'bayrağını' translates to 'his/her flag', and it signifies the object being carried.
Bayrağını onurla taşıyan sporcu ödül kazandı.
The athlete carrying their flag with pride won a prize.
The word 'bayrağını' here means 'the flag of the country' and indicates possession or association to the country.