On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$

Question

For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations:

$$x^4+16z^n=y^2 \qquad (1)$$ and $$x^4+z^n=y^2 \qquad (2)$$.

(2) is special case of Fermat Catalan and is solved.

For fixed $z$ both are elliptic curves.

Is there simple solution to (1) or (2)?

If I remember correctly, in a comment someone claimed that $x^3+z^n=y^2$ doesn't have any solution for $n>12$ via elementary means.