9to5Mac's Seth Weintraub replied that I should also check my “twitter icon cache.” It was time to dig.
Twitter operates inside OS X's Sandbox, which means all of its application data lives inside your [home] / Library / Containers folder. For Twitter specifically, that's:
[home] / Library / Containers / com.twitter.twitter-mac
Doing a quick “Get Info” from the Finder's File menu I noticed that folder was taking up almost 13GB (yes, gigabytes!) of data. I dug deeper and found the offending folder to be Data / Library / Caches / com.atebits.tweetie.profile-images. It contained over 169,000 files for 12.79GB. That's a lot of data to cache. Would I ever need all that at saved on my Mac as opposed to re-downloading it? I mean… Twitter doesn't even work if I'm not connected to the Internet.
I quit Twitter and then felt safe deleting this folder… and immediately emptying the trash. Once I did that (it took nearly 5 minutes because of all the tiny files), I relaunched Twitter and… all was well.
You might want to do the same.
Edit: Seems at least some programmer(s) at Twitter know this is an issue because, as reader Paul points out in the comments below, there is a Twitter > Empty Cache command right there in the menu bar. When you run it the command asks, “Empty Cache? This will clear and rebuild username autocomplete.” We've tested it a few times here, though, and in addition to clearing out autocomplete, it definitely does also clear out the aforementioned image cache (by deleting the folder just as above). This is a much simpler way of doing it, so long as you don't mind losing autocomplete data, too (it'll likely rebuild that just fine, though). Thanks, Paul!