The best picks globally to counter Ornn include Zeri, Aphelios, and Jinx. These champions are all good at countering Ornn with decent win rates over 56%. Even if these Ornn counter picks are not taken in your lane, it's not smart to choose him into any of these matchups. They will very likely crush Ornn.
Conversely, if the enemy team picks either Tahm Kench or Ziggs, Ornn mains should strongly think about choosing him, as he easily wins against these champions with win rates over 54% against each ofthem.