Note that some smaller lathes have quite small motors and it is best not to flog them. The Unimat SL for instance has a motor rated (not continuously) but at 80%, this means 8mins on load and 2mins cooling.
Now, give yourself the maximum advantage and ensure your tools are sharp. When turning small items like spindles a blunt tool will just push the piece away rather than cut. A small fine grinding wheel with arbour can be mounted in the lathe to sharpen the tool, but I suggest you practice on an old one or a dead drill first to get the feel of it. Place newspaper under the wheel to stop hot grit welding to the bed rail.