First — almost everyone feels this at the start, and almost no one admits it. So you're in good and quiet company.
The honest truth is that people notice you far less than you imagine. Most aren't looking, and the few who are are usually just other runners who remember exactly what starting felt like. The story in your head ("everyone can see I'm struggling") is real to you and invisible to them.
A few things help: run a quiet route or time of day at first, run with someone, or join a beginner group where everyone's in the same boat. The self-consciousness fades fast once running becomes something you do rather than something you're nervous about. Mostly it dissolves with repetition — every run makes the next one feel more normal.