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Many users currently cannot change their profile picture on Twitter. Usually, the option resides under Settings - Profile, but right now it simply doesn't exist.
The change is due to an issue Twitter is working on; Twitter has removed the option to change the background and the profile picture while it resolves the problem.
Hopefully, Twitter will restore this option soon. With Halloween around the corner, many users might want to change their picture into something more scary, and Twitter itself might soon be overwhelmed with reports from users who want to change their profile picture.