When you turn right at a crossroads the rules are the same for when you do a normal right turn.
You come into the centre of the road, without crossing the white line
You stop level with the white line of the road that you are turning into
And once you have a safe gap you go across.
The same is true at crossroads, the only difference is, if you can't find a safe gap, then you may have to wait until the lights go red to cross.
If you've already crossed the lights on green, then it is acceptable to wait for the lights to change so you can cross safely