Here's a great demo if you want to see pure speed: https://medium.com/2dimensions/showcasing-the-power-of-flutt...
Another new example of Flutter in action is Reflectly, which was recently featured on the Apple appstore: https://medium.com/reflectly-engineering/reflectly-from-reac....
In addition to flutter's own showcase you can also view flutter apps at https://itsallwidgets.com/
I've found that if you are using an older phone then you will certainly notice that Flutter is not as smooth as a native app. However, I know the Flutter team have made a lot of progress with performance so it may not be so bad.
Three largish apps you can try are:
1. The Alibaba app:
Alibaba app for Android → http://bit.ly/2NQJcMX Alibaba app for iOS → https://apple.co/2NjVGxf
2. The Hamilton (musical) app - search Hamilton on Play Strore or App Store
3. The inKino movie database app. The source code is available here: https://github.com/roughike/inKino
You may find some here: https://flutter.io/showcase/
Stats and benchmarks are neat but I'd really want to try it firsthand to know that it's "Fast".