Has she ever surprised herself and peed while she was sitting there? I have a picture of my little girl one of the first few times she actually peed in the potty. She popped up, turned around and looked (face down to the seat practically) to see the 3 or 4 little drops... lol.
Anyway, what I found worked the best with my impatient son, was to give him books to read. Sure, they can't READ yet, but they can look at the pictures and turn the pages. And pretend, especially if it is a book they know. Or you can sit on the side of the bathtub and read TO them.
Give it a try, and if she accidentally goes while she is sitting there, she will quickly make some connections. My daughter was trained by age 2. So it is entirely possible for your daughter also...
The best way to avoid the public bathrooms is 1) go before you leave home, 2) go again if it has taken more than 5 minutes to get everything rounded up to get into the car, 3) don't go to Walmart---my child ALWAYS needed to go when we were standing in the check-out line.
ETA: Start paying attention to meal and snack times. After she has had something to drink (like with lunch, not just a "sip" of something....actually drinking a few ounces), then about 10 minutes after that, MAYBE 15 minutes, ask her if she would like to try to pee in the potty. Take a book. Help her sit, and start reading her a story. I also have heard (and used it for bashful bladders) that running the faucet from the sink can stimulate urination). So maybe turn the faucet on slowly, and sit down on the bathtub side with the little book (maybe a book about using the potty?)... and read for a few minutes. No more than 5 or 6 minutes though. If nothing happens... "we'll try again later" (with a happy face). And then follow through. Maybe 15 minutes later, ask her if she wants to try again. Using this method, unless she goes while not on it, she is virtually guaranteed to have a success. THAT is when you go wild with excitement and praise and oohh look what you did! woo hoo! ;)
She will LEARN what that feeling just beforehand feels like, and start figuring out WHEN she needs to go.
Good luck.