How to deal with the trapped air in a loft attic?
First off the bathroom should be sealed from the attic space, any venting from the bathroom should be to outside.  Venting moist bathroom air into the attic space is a recipe for mold and should never happen, fix that first.

Next the attic is supposed to get hot, this keeps the heat away from the living space, if you are using the attic as living space it needs to be properly insulated and ventilated.

Don't skip the insulation step keeping the heat out is a great first step.

As for ventilation you can either as you said, add a vent fan to send the hot attic air outside, and setup other attic vents to either draw air in from outside (normal for an attic space) or draw air up from the living space (not normal, but might be what you want as you are using the attic as living space).  

Another option would be to setup interior fans to increase the air exchange between the attic and the other living space, allowing the living space HVAC to cool the warm attic air.

A big thing to consider here is that cooling the attic space with living space air is likely to make the other living spaces warmer in the summer.