The only time I've noticed a shutter delay when using strobes when there is trouble with the camera getting focus because of dim light.
There is also, sometimes a longer than "normal" blackout because of TTL metering.
It may also be because you have the camera set to wireless for flash. I've found that only is useful when using my Nisson Air speedlites. They emulate the Sony optical wireless system.
With Godox X system it has to be set to fill flash, slow, or second curtain. If set to the last two shutter speeds are much slower. Fill flash is the default.
Having used the Godox X system now for literally thousands of TTL and manual exposures I can't think of more than 4 or 5 times where the shutter was appreciably slower. As mentioned in each case there was some external cause.