Let $a$, $b$ and $x$ be curves on $S$ such that

• $a$ and $x$ intersect once,
• $b$ and $x$ intersect once, and
• $a$ and $b$ have a large algebraic intersection number.

(I assume that $S$ is orientable.) Then there is a surface $\Sigma_a$ connecting $x$ and $x+2a$ with $\chi(\Sigma_a) = -1$ and a similar surface $\Sigma_b$ connecting $x$ and $x+2b$. Combining $\Sigma_a$ and $\Sigma_b$ we get a surface of Euler characteristic $-2$ connecting $a$ and $b$. Thus the geometric intersection number of $a$ and $b$ does not give a lower bound on the complexity of a surface joining them.

(In your original question you ask if $\chi(\Sigma) \le n$, but $n$ is positive and $\chi(\Sigma)$ is negative, so I assume you meant $|\chi(\Sigma)|$.)

Let $a$, $b$ and $x$ be curves on $S$ such that

• $a$ and $x$ intersect once,
• $b$ and $x$ intersect once, and
• $a$ and $b$ have a large algebraic intersection number.

(I assume that $S$ is orientable.) Then there is a surface $\Sigma_a$ connecting $x$ and $x+2a$ with $\chi(\Sigma_a) = -1$ and a similar surface $\Sigma_b$ connecting $x$ and $x+2b$. Combining $\Sigma_a$ and $\Sigma_b$ we get a surface of Euler characteristic $-2$ connecting $x+2a$ and $x+2b$. Thus the geometric intersection number of two curves does not give a lower bound on the complexity of a surface joining them.

(In your original question you ask if $\chi(\Sigma) \le n$, but $n$ is positive and $\chi(\Sigma)$ is negative, so I assume you meant $|\chi(\Sigma)|$.)