If this girl at the gym is constantly looking at you, there is a good chance that she likes you.
She may keep looking away whenever you try to look back at her because she is very shy and she doesn’t want you to catch her looking at you.
She always looks at you with a straight face and then looks straight down when you catch her looking because she may be worried that if she is more expressive with her face, you will be able to tell that she is attracted to you.
Being that she isn’t sure whether you are attracted to her or whether you are already in a relationship, she may not want to be too obvious with her attraction towards you.
Being too obvious may make her feel awkward and self-conscious.
It is unlikely that she thinks you are weird.
If she did, she would do everything in her power to avoid looking at you so that you don’t get the wrong idea and then feel emboldened to approach her as a result.
In your case, she has been looking.
The both of you have made eye contact a few times and she even smiled at you briefly on one of those occasions.
The way she has been behaving would indicate that she is attracted to you.