There are kids-friendly shopping malls in Singapore. For one, Westgate has a toilet specially for kids. You could bring her to watch a movie over at JEM and let her use the toilet at Westgate. Otherwise, you could let her use the ladies but do keep a close eye on her. Try to approach a mother with kids on her way to the ladies and ask if she can keep an eye on your daughter. http://bumblebeemum.net/2015/05/28/top-5-kids-friendly-shopping-malls-in-singapore/
It depends on how independent your 4 year old is. My ex husband would bring my daughter to the handicap toilet and he'd start guard outside while she goes in. Afterwards he'd ask the usual "Have you wash your hands" and that's it. Worse case scenario, he brings my daughter into the men's toilet while closing her eyes and start guard outside the cubicle. He never goes into the cubicle with her at all.
Most of the theatres nowadays have restrooms with attendants there. You can request the person to see your daughter through. Also there are restrooms for disable people, she can use that too. But if you go often with your daughter, it is better to train her so that you can go out without worrying much about her pee time.
You can let your daugther to use handicapped toilet and all shopping malls have this facility. A 4 years old kid should be able to use the toilet seating (just make sure you wipe the seat cover before hand using sanitiser). You can wait for your daugther outside of handicapped toilet.
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thanks