Are there ways to correct high palate and narrow jaw without surgery?

Depends on age. If this is a concern for one of your children, then yes. Routine palatal expansion (if needed) can be done quite easily. If you are an adult, it depends on the amount of expansion that you may need, as there are limitations to what can be done non-surgically and still not cause harm to other structures in the mouth. Speak to an orthodontist in order to have the proper evaluation.
Can't agree. There may be ways to widen and expand the arches non-surgically. Find someone trained in neuromuscular orthodontics and functional orthodontics.

Related Questions

How to correct high palate and narrow jaw without surgery?

Palatal Expander. Many types of fix and removable orthodontic devises called palatal expanders can be quite effective at correcting a high palate/narrow jaw condition particularly in a growing individual. Read more...
Surgery. At your age, oral surgery only option. In a growing child, palatal expansion device often helpful. See orthodontist for options. Read more...