Super Saiyan said:
Why was he so arrogant after being cucked by Mihawk? Mihawk could've ended his life then and there, but suddenly Don Krieg thought he could conquer the world after he left
That was kind of the point of his character. Krieg's designed to be the kind of guy who's resigned himself to trying to conquer low hanging fruit in East Blue to be the big fish in a small pond after being proven he can't take on the Grand Line. This makes him a decent foil for Luffy's ideology of wanting to strive to the top and take on the big names of the Grand Line rather than run away from them, making it essential Luffy were capable of taking on someone who'd faced the Grand Line and lost.
This seems to intentionally be a running theme amongst East Blue's main villains - Buggy was the weak link of a Legendary crew and had ties to Shanks to make him a necessary stepping stone, Kuro represented complacency and fatigue compared to Luffy's ambition, Krieg reserved himself to taking on something he had confidence in controlling rather than striving forward and Arlong was the same but with more bragging rights by having actually come from the Grand Line. That's not to say the East Blue villains are great or anything (unless comparing them to the disappointment Kaido and his crew have become), but they do have a clear narrative purpose to their goals of intentionally targeting small fry and acting as though they're hot shit by limiting themselves to easy targets.
Kenshi said:
Arlong's ambition being to merely rule a backwater like East Blue when he was allegedly once on Jimbei's level is
as well. People who had worse feats have gone on to enter the New World. Why the fuck is he so small time?
To be fair, most later translations (as well as some level of retconning in Part 2) show Yosaku was referring to their rank rather than power.