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Fix 
$d\geqslant 2$ and a field 
$k$ such that 
$\operatorname{char}k\nmid d$. Assume that 
$k$ contains the 
$d$th roots of 
$1$. Then the irreducible components of the curves over 
$k$ parameterizing preperiodic points of polynomials of the form 
$z^{d}+c$ are geometrically irreducible and have gonality tending to 
$\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of 
$z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.
