Assume Vojta’s Conjecture. Suppose \(a, b,\alpha,\beta\in \mathbb{Z}\), and \(f(x),g(x)\in\mathbb{Z}[x]\) are polynomials of degree \(d\ge 2\). Assume that the sequence \((f^{\circ n}(a), g^{\circ n}(b))_n\) is generic and \(\alpha,\beta\) are not exceptional for \(f,g\) respectively, we prove that for each given \(\varepsilon >0\), there exists a constant \(C = C(\varepsilon,a,b,\alpha,\beta,f,g)>0\), such that for all \(n\ge 1\), we have \(\gcd(f^{\circ n}(a)-\alpha, g^{\circ n}(b) -\beta) \le C\cdot\exp({\varepsilon\cdot d^n}).\)