My EM1 is set to sync flash at 1/320 sec (Menu F X-sync and Slow Sync). This works with the included flash, FL-LM2, but not when I mount my Godox TT350o. When the speedlight is mounted I can only do 1/250 sec. I would like to go faster, mostly to shoot macro and things like flowers to controlthe background light relative to the subject.
Anyway, is my 1/250 sec limitation due to the type of flash, or am I doing something wrong?
The limitation is due to the focal plane shutter in your camera.

It's a camera body limit. A focal plane shutter uses two curtains to control shutter speed. The shutter speed you set determines the length of the gap between the two curtains. At 1/250s (or 1/320s, depending on the flash/camera communication), that gap is just big enough to uncover the entire sensor during the flash burst. But any faster than that, and the gap will be smaller than your sensor, and part of the sensor will be covered by one or both curtains, and you'll get black/dark bands at the top and/or bottom of the frame because when the flash went off, the whole sensor wasn't exposed to the light.
Most focal plane digital cameras, however, can use flash with faster shutter speeds by using an HSS (high-speed sync) flash mode; this is also called FP (focal plane) flash. This pulses the flash continuously during the entire duration of the exposure, so the flash basically acts more like a continuous light source. But. The repeated pulsing required to do this uses a lot more power than a single (non-HSS) flash burst. With macro subject distances, this hopefully won't be an issue, but for portrait shooting, you might be fighting to get enough light from the flash [see
Neil van Nierkerk's HSS tutorial].
The EM1 can do HSS. You just have to set it on the flash. On the TT350-O, press the SYNC button until you see an H with a lightning bolt icon. That means HSS is turned on. With an FL600R or FL360L, you'd be using either FP TTL Auto or FP Manual mode to get HSS.
With HSS enabled, I'm able to sync all the way up to 1/8000s with my GX7 without any banding. However. The power settings on the TT350-O, while in HSS, are limited to full to 1/16 power. You can't go any lower.