If you look at the request it does, use "copy as curl", replace the word with "$word", and put it between:
xclip -out -selection clipboard | while read word; do <the curl command>; done
Then you can probably do the whole list at once.
There is probably also a javascript way to automate this, but if they do it with ajax, this might be faster. Since I do this sort of automation regularly, I aliased the clipboard pasting to "c-v"[1] and so it's quite easy for me to do this sort of thing quickly without having to remember the exact flags and type all of that.
[1] .bashrc addition: alias c-v='xclip -out -selection clipboard'
If you feel like trying something else than xclip, xsel allows you to easily remember the command with just `xsel -o` instead of that.
Also, if the JS way would use AJAX, it would take the exact same time to perform the requests via the browser, as it would with curl, unless there is some particular quirk with their server/setup/deployment/god, or if you have a computer that doesn't run Twitter (the client side application) so good (bottleneck in cpu/ram rather than network speed)
Wondering if the downvote came because of someone thinking xsel is inferior to xclip, or if there is a misunderstanding what AJAX really is... Would be great if people say something more than "I DIDNT LIKE THAT"