The depressing thing for me is to know that when they show me a Starbucks ad, Twitter makes pennies for that. I would much rather just pay those pennies myself and not see ads.
It lets you pay a specific amount per month (IIRC from $2 to $15 or so) that you use to "bid" for ads spots just like the advertisers. If you win, then you'll see whatever you choose in place of the ad. Any money left over at the end of the month is refunded.
It won't help you in the twitter case, but it does remove a good amount of ads across the net, and it even has controls that you can go in and whitelist/blacklist some websites from receiving contributions, as well as a full breakdown of where your money is going.
I've removed 16,000 ads since I signed up in april 2015