But doesn't Jackson straight up tell his lawyer that he's the bad guy just because he's white?
Not exactly. What he's saying is that, no matter his noble intentions, that the lawyer is a white guy in a specific society and will never quite understand Jackson's position or motives when discussing the crime with him. And, intentionally or not, he's part of and participates in that society. BTW, you'll remember at the end that they are having a barbecue and socializing together. If I had been a black guy growing up in a small southern town where things were stacked against me, I'm pretty sure I'd have an attitude. As a matter, of fact, I grew up in a small southern town as a white guy, but witnessed some pretty shitty things that occurred. So, I would not think of Jackson's character as "racist". Pissed off and resentful, sure. But not a racist.