Yes the Avatars repeat themselves but NOT in every Chaturyga.
In the Yog Vasishtha Chapter 22 Kak Bhushunda, an immortal crow tells Brahmarishi Vasishth of the events of significance that he has witnessed multiple times. In this he mentions many avatars of Lord Vishnu as well:
14 Thrice I have seen the tyrant Hiranyaksha levy his tax upon the gods in heaven, hurling the fruitful earth with all her healing
and medicinal plants underneath the ocean.
15 I saw Hari (Vishnu) come down six times in the shape
of Renuka’s son, Parashurama, and remove the kshatriya warrior caste for very long periods.
16 O sage, I remember the return of a hundred Kali Yuga ages and a hundred incarnations of Hari in the form of Buddha and as the son of royal Suka or Suddhadana in the land of Kirata.
He also narrates having witnessed the Narsimha Avatar 3 times, Ramayan 11 times and Mahabharat 16 times:
30 I remember Lord Vishnu descending many times on earth to destroy ferocious rakshasa demons, and is now to appear here the eleventh time under the name of Rama.
31 I know Lord Hari has come down three times in his form of the half lion half man Narasimha to thrash the demon Hiranyakashipu as many times like a lion killing an elephant.
32 Vishnu is yet to be born in his sixteenth incarnation at Vasudeva’s abode for the purpose of rescuing the earth from the burden of the oppression of its tyrant lords and despots.
If the Avatars were repeated in each Chatur-yuga the number of these repetitions mentioned by Bhushunda would be in thousands as he had been around for many Kalpas. This is mentioned in Chapter 20 of the same book:
1 Bhushunda continued:—
This world has existed in the previous kalpa in the very same state as it does at present. There is no
variation in the formation or location of anything in any way.
2 Therefore, O great sage, I am
accustomed to look to the past and present with an equal eye. I will relate the events of my past life
and bygone ages for your information as if they exist with me even now.
All these lead us to believe that even though the major Avatars are repeated in different Manvantars and Kalpas, the low numbers of repetition imply that they are NOT repeated in every Chaturyuga.