“I didn't want to make her feel self-conscious.”
Today your daughter made a really funny mistake on her science homework. You wanted to laugh about it, but you didn't want to upset her so you stopped yourself from laughing. Now you're telling your husband about it and you offer him this explanation.
I didn't want to make her feel self-conscious.
The phrase "make (someone) feel ___" is most often used with emotions like these:
I didn't mean to make you feel uncomfortable.
Does that make you feel unwelcome?
These are all emotions that involve sticking out from a group of people.
"Make (someone) feel ___" doesn't work as well with emotions like "angry" or "sad". Instead, it's normal to say:
Doesn't that make you angry?
To "feel self-conscious" means to worry a lot about what other people think of you. In the example above, if the speaker had laughed at her daughter, her daughter would have felt self-conscious about her mistake.
As another example, imagine that you're at a party where you don't know many people, and you're dressed up a lot more formally than the other people there. You can describe your feeling as:
I felt extremely self-conscious the whole night.