There are several synonyms for the phrase "turn thumbs down on." One similar expression is to "veto" something, which means to reject or prohibit it from moving forward. Another synonym is to "disapprove." This means that you do not approve or agree with something. A third synonym is to "reject," which also means to turn something down or refuse it. One other synonym is to "deny." This means to refuse to grant something or to reject a request or proposal. Overall, "turn thumbs down on" can be replaced by several other phrases that convey the same meaning of rejecting something.