Typically in deckbuilding, the beam is sized according to the post (e.g. a double 2x10 beam with a 6x6 post). For building enclosures that support roofs or shade structures where it supports less weight than deck + people, the post width and beam width can be much closer together, like a 4x4 post and 4x6 beam.
In all these cases though, typically either the post is notched to support the beam (when the post is thicker than the beam) or the beam is supported by a "post cap" bracket which provides a secure connection between post and beam members.
The reason I talk about "typically" is because the post cap brackets are sized for each situation and it's easy to find caps for 4x4 posts carrying 4x__ beams, and 6x6 posts carrying double-2x__ beams. I wasn't able to find a post cap for your sizes, but that doesn't mean it doesn't exist. The closest I found for your situation is a single-side 4x4 post to 4x__ beam that is meant to be used in pairs to sandwich the beam on top of the post.
In your edit you mention maybe needing to replace the 4x4 post with a 4x6 post; that sounds like a good start so that the post is at least as thick as the beam it's supporting. You may be able to use a 4x4 or 6x6 post cap bracket that has triple-2x__ support on top, most of them only attach to two sides of the post so it depends on which sides of your posts (the face under the beam or the same face as the beam like the bracket picture above).