Making a name for oneself in the world can be challenging, but there are various synonyms to use instead of the phrase "make way in the world." One such synonym is "establish oneself," which means to successfully achieve a reputation or position in society. Another synonym is "forge ahead," implying that one is bravely pushing forward despite any obstacles in their way. "Carve out a niche" is another synonym that suggests one has found a unique and profitable area of expertise. "Blaze a trail" can mean breaking new ground or paving the way for others to follow. Lastly, "make a mark" tells that one has left a lasting impression on the world.