> (It's hard to see a bike at speed going straight when a car is turning right.)

This is addressed by having a "no turn on red" sign as I suggested: you don't have to see the bicyclist at all because the bicyclist shouldn't be in the intersection at all when you're turning with a green turn arrow.

Why is a bike going at speed when the light is red? Shouldn't the bike also stop at the red light?

The bike isn't going at speed when the light is red (at least, if they're following the law). This is when the green turn arrow should be lit: when bikes aren't moving.