This panel is NOT about comparing Mihawk & Newgate power wise. It's establishing the themes of arc and Mihawk's character.
This panel is frequently cited as Mihawk admitting inferiority to Newgate. I disagree. I disagree with that reading of the panel. Whether you believe Mihawk is actually stronger or not is a different story but this panel shouldn't be used in your argumentation. But if this panel isn't about powerscaling then what is it about? The panel is establishing one of the themes of the arc. That theme in specific is about the strength that comes from those who can attract allies. This is important and the reason that Mihawk is the one used to illustrate this is because this is eventual set up for Mihawk himself to understand why Zoro wants to be taught by him as well as his eventual decision to join Cross Guild.
Mihawk is a man who is defined by his strength and his loneliness at the top. He lost his best rival in Shanks when he sacrificed his arm for Luffy's future. This event shattered Mihawk. Shanks was the closest swordsman to him in the world and was his best challenge in his career. The duels the two shared rocked the New World and had big names like Newgate watching. In Mihawk's mind Shanks was the rival who would push him to the greatest heights and vice versa. But it didn't happen. Shanks went to the East Blue. A very weak sea and then returned with his entire arm missing. News of this fundamentally rocked the New World as one of it's rising starts seemingly suffered a debilitating injury while in said weak sea. Now while we the audience know why Shanks did this (To bet the future on Luffy as the man to inherit Roger's legacy & to sever his connection with Imu) the rest of the world does not. To the rest of the world this was absolute madness. The rivalry Mihawk & Shanks shared died that day. Even though Shanks would rise to be an Emperor and respected around the world to Mihawk it didn't matter. Shanks had now hard capped his potential in losing his arm and has now tainted the rivalry permanently. Whatever good spirited fun the two had was gone in Mihawk's eyes. Mihawk took the title of World Strongest Swordsman and nobody has seriously challenged him since. Shanks seemingly is content with this status change which probably drives a wedge between him and Mihawk more.
Fast forward to Marineford. Mihawk has already met the Strawhats and has a glimpse of the potential that the next generation has. The generation that his greatest rival bet his arm on. While Mihawk himself acknowledges the strength that comes from the belief people have in each other he doesn't see it as something that can beat him. Enter Newgate. Newgate is known as the Strongest Man in the World, the rival to the late Pirate King Gol D. Roger, & the de facto guardian to the Empty Throne that Roger left vacant with his death. Newgate is the man closest to the One Piece. If anyone can show the strength of belief in others its him, because everyone knows that Newgate's family will die for him... or so they say. With Newgate standing out front on the Moby Dick in the open with seemingly no protection he is the only target on the battlefield. A very bold choice for what will end up being the final battle of the Great Pirate Era. Mihawk notices this and steps forward. He feels the "distance" between Marine HQ and Newgate is pretty small. Is Newgate planning on running in head first to save Ace or will his Division Commanders, his sons protect him from any attacks and act as the real vanguard? The only way to find the answer is to "measure the distance". Can Mihawk attack Newgate unopposed or will he be rebuffed? Is the power of belief in others stronger than the power that comes from the solitary peak? Mihawk attacks and Jozu steps in. As Mihawk expected the "distance" was an illusion. It was to bait the Marines into seeing him as vulnerable and making a reckless play. Kuzan may have moved first by defending Marineford from the Sea Quake but Mihawk made the first attack on Newgate directly. The war is on.
Fast forward to later on and Squard has betrayed Newgate. He stabs his father figure after being deceived by Sakazuki and Sengoku. After seeing Mihawk's suspicion be correct and that the Division Commanders will defend Newgate until the end the next step of the war is to shorten the "distance" by using someone who can get past the Division Commanders. Enter Squard. Squard represents how even the strongest belief in a man can be shaken if you press the right buttons. Using Roger as the trigger Squard wavers and he attacks Newgate. This forces Newgate to enter the battlefield himself and lead himself. The "distance" has been shortened. Newgate is now more open to attacks than before and the Marines know it.
Fast forward to Luffy & the Impel Down causing havoc across the battlefield and Mihawk now has his second chance to test whether the belief in others truly can match his strength. He has seen glimpses of it in Luffy. He saw Zoro pledge himself to Luffy. He saw how Luffy got Baratie to be on his side and then defeated Don Krieg. He read the papers. He knows this kid is the one Shanks bet their rivalry on and now he personally wants to see if Shanks was right to do so. So he attacks him. Mihawk has the clear advantage and it would seem Shanks would be wrong. Until he's not. Marco commands Vista to protect Luffy from Mihawk. Vista does so and suddenly the "distance" between Luffy & Mihawk widened. That's when Mihawk sees it. The most dangerous power on the seas. The power that can surpass him. The ability Luffy has to get others to believe in him and support him. It is then that Mihawk realizes that Shanks is right.
Fast forward to the end of the war. While many meme about how Mihawk left when Shanks arrived and use it to say that Mihawk didn't want smoke I interpret this part differently. I think Mihawk doesn't want to acknowledge to Shanks' face that he was right. That Shanks' bet was correct and that he was wrong to have doubted him. My reasoning for this is that once he returns to Zoro he has now changed his mind about training him. Because despite his loneliness and strength even Luffy's power swayed him. Whether he fully accepts it or not he too has some belief in Luffy. The "distance" between them is now larger than ever. And with that in mind he now sees that Zoro's strength has the potential to surpass him. Not just in talent but now because he knows the man Zoro is willing to die for. He trains Zoro because he knows that Zoro's belief in Luffy will push him to eventually be a worthy challenger. A challenger that may surpass him & Shanks. He sees the vision.
Fast forward to present and Mihawk now is part of Cross Guild. He has foregone his previous lonely nature and is now part of something bigger. Despite his claims that he just wants peace of mind he now is part of a crew. He is now competing with others for the One Piece. He is now in an environment where Cross Guild's belief in Buggy has now put them on the offense. They are going to war with the Marines, they are fighting for the One Piece and most importantly of all Mihawk is getting closer to the challenge he has so dearly missed this whole time.
In conclusion, the panel above is not about powerscaling. It is about Mihawk, his character, one of the biggest themes that surround his character arc and ultimately about where his character will go in the future. If you think Mihawk was stronger than old man Newgate then this might fall in line with how you read the story. If you think old man Newgate is stronger than Mihawk this is not a debunk of that thought. This is more of a debunk of a specific argument used in thinking thbucks.
TLDR: Read the post you illiterate fucks.