くらいなら死んだほうがましだ

I'd rather die than

→ くらい
phrase Rare #37,867
I'd rather die than Used to express an extreme preference for death over doing something undesirable. Usually hyperbolic rather than a literal statement. The part before くらいなら describes the alternative being rejected.
JA: こんな恥ずかしい思いをするくらいなら死んだほうがましだ。
EN: I'd rather die than go through something this embarrassing.

自分の信念を隠すくらいなら死んだほうがましだ

I’d rather die than hide my beliefs.

そんなことするくらいなら死んだほうがましだ

I'd rather die than do something like that.

彼は毎朝早く起きるくらいなら死んだほうがましだ

He’d rather die than get up early every morning.