We investigate Bak-Sneppen coevolution models on scale-free networks with various degree exponents $\gamma$ including random networks. For $\gamma >3$, the critical fitness value $f_c$ approaches to a nonzero finite value in the limit $N \to \infty$, whereas $f_c$ approaches to zero as $2<\gamma \le 3$. These results are explained by showing analytically $f_c(N) \simeq A/<(k+1)^2>_N$ on the networks with size $N$. The avalanche size distribution $P(s)$ shows the normal power-law behavior for $\gamma >3$. In contrast, $P(s)$ for $2 <\gamma \le 3$ has two power-law regimes. One is a short regime for small $s$ with a large exponent $\tau_1$ and the other is a long regime for large $s$ with a small exponent $\tau_2$ ($\tau_1 > \tau_2$). The origin of the two power-regimes is explained by the dynamics on an artificially-made star-linked network.