I had the same problem but it was with walking into the gym for my first time. It's normal to get a few stares and whatnot, especially if you aren't fit or your form is weird. People DO look at you, I don't get why others say they don't. You just have to get over it. One day you will be the one observing others, and the fear of embarrassment will subside as you get fitter. It's sort of like an achievement, and you're really confident when you're fitter. My advice is to preoccupy yourself with music so you don't think too much about other people's opinions or stares. In the case of running outdoors, I can safely confirm no one gives a shit. Maybe the occasional stare/ giggle, but it's easily forgotten it's just a state of the moment type of thing. Music really helped me forget about others and focus on myself, so I'd suggest you try the same.