Let $X,Y$ be normed space. Assume there exists $c_1 , c_2>0$ such that $$ c_1 \| x \|_X \leqslant \| x \|_Y \leqslant c_2 \| x \|_X. $$ Then if $ \| x_1 \|_Y \leqslant \| x_2 \|_Y$ then $\|x_1 \|_X \leqslant \|x_2 \|_X$ holds? I think this does not hold by using only the definition of the equivalence of norms.
No, of course it doesn't hold. In fact your property would imply that there is some constant $c$ uch that $\|x\|_X = c \|x\|_Y$ for all $x$.