I read today on not-implemented.com :
Sha-256 should be chosen in most cases where a high speed hash function is desired. It is considered secure with no
On your side, the password hash needs to be computed rather rarely. But an attacker who tries to brute force a password from a stolen hash, relies on computing as many hashes as possible.
So, if your login now takes 100 ms instead of 0.1 (probably less) that's not really a problem for you. But it makes a huge difference for an attacker if he needs 2000 days to break a password instead of 2 days.
bcrypt is designed to be slow and not to allow any shortcut.