Q:

Does traffic going from AWS US-East to AWS Singapore travel over a priority route? Or is it subject to normal routing rules?

Does traffic traveling from AWS US-East to AWS Singapore travel over a priority route?  Or is it subject to normal internet routing?
To paint an example:

Server A is in AWS-US-East (us-east) 
Server B is in AWS-Singapore(southeast-asia) 
Server C is in Azure-Singapore (southeast-asia)

Assuming the same network request is made to two different availability zones:
A <-- Http Request 1 --> B
A <-- Http Request 2 --> C

Would there be a noticeable speed increase when making network requests between Http Request 1 and Http Request 2 given that Request 1 travels between two AWS availability zones?

A:

See Behind the Scenes: Exploring the AWS Global Network (NET305) from AWS re:Invent 2018, starting around 25:00.  AWS manages the transport infrastructure connecting all the regions together, except for China... so inter-region traffic is routed across high-bandwidth, uncongested links under AWS's control.
This should typically offer the best case performance across distances like these.  Connections on the public Internet are not likely to be any better.  They might be comparable or they might be worse, but it is impossible to definitively say what the anticipated differences between your specific scenarios might be -- or how performance on the Internet might vary over time.  Over the AWS network, performance should be quite consistent over time.
If you find a significant difference, you can deploy a proxy server in EC2 in ap-southeast-1 (Singapore) and use it from us-east-1 to contact the external service, instead of contacting it directly via the Internet.  This will cause you to pay cross-region as well as out-to-Internet pricing for bandwidth, but that may be worthwhile if the route is either superior in latency or more consistent in performance and stability over time.

