General encouragement:
* Inspire
* Motivate
* Stimulate
* Support
* Boost
* Promote
* Foster
* Empower
* Uphold
* Back
* Advocate for
More specific to confidence:
* Bolster
* Strengthen
* Empower
* Embolden
* Hearten
* Cheer
More specific to action:
* Urge
* Prompt
* Incite
* Goad
* Impel
More specific to positive growth:
* Nurture
* Cultivate
* Develop
* Nourish
The best synonym will depend on the specific context. Consider the nuance you want to convey and choose the word that fits most appropriately.