Those pros you see are likely using DSLR cameras. They can probably generate apertures your camera isn’t capable of to let in more light, and they are capable of giving decent results at much higher ISO. Some also have stabilized lenses or sensors. ISO1600 at f1.4 with a stabilized lens is going to give good indoor capability for non-moving subjects and even capture limited movement if there is enough light.
There is no magic in manually setting shutter speed. Long before the light is a real problem the lens is open all the way in auto or program to generate as much shutter speed as possible for the available light. Speeding up the shutter is just going to give dark pictures. And if you bring those dark pictures up in an image editor they will be extremely noisy.
One thing you can do is to not zoom. Most consumer cameras let in about three times the light at wide as they do at full zoom. You can also handhold at a lower shutter speed without blur at wide angle.
Stabilization helps, but only for subjects that aren’t moving. ISO 400 plus stabilization will give some pretty good limited light shots if the subject isn’t moving. There are some stabilized cameras that are quite good at ISO400 like the Canon SD 700 & 800 and the Sony T10/30/50. Even with stabilization you are better with wide angle in limited light.
If your camera has a hot shoe or sync connector, get an inexpensive automatic flash attachment. Either bounce flash or a diffuser gives excellent light. Even if you can’t add a flash attachment it is often better to just turn the flash on than to get blurry noisy photos.