Potty on the Go or "On The Go Potty" products are GREAT! We have those in our van and my Mom's car. THIS is a great thing... all kids go through aversion to public toilets. It WILL pass. It's a phase.
In the meantime, our girl would go in the portable folding potty in our car. No problem. No anguish. No hassle. She can still use it if a public place is not available or just very unsanitary and stink, or if we are on the road & there is just no toilet place anywhere close by,we just pull over and she can pee. And she is 5 years old now. It's great for emergencies.
It's a product that lasts a long time, no matter what age, as you can see. We had ours since our girl was 2 years old.
I recommend it. There are many different brands... on Amazon.com
You can even carry it in a tote bag with you if you want... which all Moms usually lug around anyway, since we have kids. It's not a bulky thing. We LOVE that we got ours. It has been a life saver in MANY situations.
My girl, at about 2 years old, was literally afraid of public toilets... the smell, the noise, and the crowded stall. I never "forced" her or punished her or made her go. (I myself didn't even want to go in some of these stink messy public toilets). It's not big deal... they grow out of it. No worries. LOTS of kids go through this. My feeling is, why force it and make them then they have an accident in the middle of the mall... this is worse AND it embarrasses them, and what a mess to clean if in a store or in the car!
Good luck,
Susan