By hypothesis there is a bijection $h:A\cup C\to B\cup C$, $A\cap C=B\cap C=\varnothing$, and you want to show that there is a bijection from $A$ to $B$. If you were lucky enough to find that $h[C]=C$, you’d be done, because then $h[A]$ would have to be $B$, and $h\upharpoonright A$, the restriction of $h$ to $A$, would be a bijection from $A$ to $B$. Unfortunately, there’s no reason to hope that $h[C]=C$. If $C$ is infinite, $h[C]$ might be a proper subset of $C$, and then we could easily have $|A|>|B|$.
In fact, if $C$ is infinite the result is false; here’s a counterexample. Let $C=\Bbb Z^+$, the set of positive integers, $A=\{0\}$, and $B=\Bbb Z\setminus\Bbb Z^+$, the set consisting of $0$ and the negative integers. Then $A\cup C=\Bbb N$, $B\cup C=\Bbb Z$, and
$h:\Bbb N\to\Bbb Z:n\mapsto\begin{cases} \frac{n}2,&\text{if }n\text{ is even}\\ -\frac{n+1}2,&\text{if }n\text{ is odd} \end{cases}$
is a bijection from $A\cup C$ to $B\cup C$. Obviously, however, $A$ and $B$ do not have the same cardinality.