I wouldn't say there's an actual reason, but before Ilwoo came around, he didn't really have anyone. He was alone, and then there was the ugly bastard. It's clear that this is emotional, mental, and psychological manipulation. He knows exactly what he's doing and has used Bakhee's situation to his advantage. His dad is in jail, and when he gets out, Bakhee knows he wants to hurt him. So he's basically manipulated him into believing that no one is going to have his back and that no one is going to be able to save him from his dad.
That's why he keeps pushing the idea that he can move him somewhere far away, provide for both of them, and basically keep him safe from everything he's afraid of. He has the money and connections to actually do that, while Ilwoo is someone Bakhee just met who doesn't even know the full situation or how serious it is. So in Bakhee's mind, he feels like he can't just run away from the person who's offering him protection when he doesn't even know how to tell Ilwoo, "My dad wants to hurt me when he gets out of jail, and that's why I want to leave so badly."
And that's also why I don't think it's as simple as Bakhee just choosing to stay with him because he wants to or because he literally can't get away from him. It's psychological. He's been manipulated into believing that this person is the only one who can protect him.
does anyone knows why does the black hair mc can't get away with that gray hair dude?? I'm skimming through recent chapters. I don't want to read this yet, I'm just curious