  Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist
family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of
the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we
prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many
integers $n > 0$ or $g(K_n) / g_4(K_n) \to 1$ as $n \to \infty$, then either
the winding number of $K$ about $c$ is zero or the winding number equals the
wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family
$\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the
latter must occur. We further develop this to show that $c$ is a braid axis of
$K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain
infinitely many tight fibered knots. We also give a necessary and sufficient
condition for $\{ K_n \}$ to contain infinitely many L-space knots, and show
(modulo a conjecture) that satellite L-space knots are braided satellites.
