  In this paper, we prove under mild hypotheses the Iwasawa main conjectures of
Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes.
Our proof is based on the study of the two-variable analogues of these
conjectures formulated by B\"uy\"ukboduk--Lei for imaginary quadratic fields in
which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our
results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in
analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm
GL}_2$-type for non-ordinary primes $p>2$.
