Does an accelerated charge always radiate? For example the current electrons in an electric circuit when moving through a turn they are accelerated, do they radiate because of that acceleration?
An individual electron going around the loop would of course radiate. However, if we have a constant current going around the loop, there are many electrons and their individual radiations cancel each other, so there is no net radiation. They balance each other out. We can show there is no net radiation by solving Maxwell's equations on the net current, which is constant. See, e.g., McDonald's Why Doesn’t a Steady Current Loop Radiate? (pdf).
There would be some transient radiation when you change the current. Also, in the real world, there would be some infinitesimal radiation due to the fact that the current won't truly be constant at the microscopic level, i.e. because of Johnson noise etc.
Generally speaking, accelerated charges radiate. For the case of a current in a wire, the acceleration is so small that the radiation is hard or impossible to detect.
For example, consider 1 amp traveling down a wire of radius 1mm. How fast do the electrons move? To solve this we need to know the number of mobile charge carriers per unit volume and a few constants. The answer is $v = $ 2 meters per day. You can convert this speed into an acceleration using the formula $a = v^2/R$ where $R$ is the radius of curvature of the corner. It is a very small number.
For comparison, the radiation that makes it so difficult to speed up electrons in a collider experiment comes from electrons that are traveling essentially at the speed of light.
