under thumb, under the thumb of

under thumb, under the thumb of は何とかの支配下にある、という意味です。4:08 Many of you might be wondering why anyone would choose a life like this, under the thumb of discriminatory laws, eating out of trash cans, sleeping under bridges, picking up seasonal jobs here and there. 差別的な法の下で生きる、ゴミ箱をあさる、乞食生活を送る、日雇い労働で生きる、、、多くの方が、誰が好き好んでこのような生活を送りたいのかとおもわれるでしょう。

https://www.youtube.com/watch?v=-vBDtP3uZRg  1:10 they were living under the thumb of a group or police force that didn't represent them very well. 自分たちが選んだわけじゃない警察や組織の支配下で生きていくのに辟易としている人々。
https://www.youtube.com/watch?v=JKqPtfm1WCE  22:35 the weight and the persistent sexual arousal syndrome have both got to come under her thumb. ジェニーは体重と持続性性喚起症候群（じぞくせいせいかんきしょうこうぐん）をコントロールしなければならない。
https://www.youtube.com/watch?v=dxLOhbEId2c  3:41 josh as a severely troubled teenager under the thumb of an abusive father.

練習問題解答
a-3782 under the thumb of
a-9052 under the thumb of
b-4207 under the thumb of