Burgeon forth is a phrase used to describe something that is expanding rapidly or flourishing. There are several synonyms for this phrase, including proliferate, sprout, bloom, grow, thrive, and prosper. Proliferate can be used when there is a rapid or excessive growth of something. Sprout is used to describe the initial growth of a plant or an idea. Bloom is used to describe the unfolding of a flower or when something is at its peak. Grow is a general term used to describe the development of something. Thrive is used when something is growing healthily and strongly. Prosper is used to describe the success and growth of a business or individual.