We study K-stability of algebraic fiber spaces. In this paper, we prove that Calabi-Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are log-twisted K-stable, where log-twisted K-stability is a variant of K-stability. Moreover, we prove that there are cscK metrics on rational elliptic surfaces that have only reduced fibers. Contents 14 4.2. Klt-trivial fibrations and log-twisted stability 20 5. Applications to Rational Elliptic Surfaces 27 5.1. Log-twisted K-stability of the base curve 27 5.2. GIT-stability in the sense of Miranda I: Weierstrass fibrations 29 5.3. GIT-stability in the sense of Miranda II: Cubic pencils 32 5.4. Applications to Miranda's conjecture on Chow stability 35 Appendix A. Some results on log-twisted Fano pairs 38 A.1. Notation 38 A.2. Ding stability of log-twisted Fano pairs and δ-invariant 40 A.3. Log-twisted K-semistable Fano pair and destabilizing boundary 44 References 49