The appearance of segmented controls has changed considerably in iOS 13. To see some examples, open the Apple Calendar app on iPad and you can see a Day/Week/Month/Year segmented control in the navigation bar (the iPhone version has a different UI). Or click on the Info button in the Maps app to bring up Maps Settings to see another example. Or create a quick test project with a segmented control in it. Experiment with putting different background colours behind it.
By default on iOS 13, the segmented control has a white "bubble" behind the selected text/image. In dark mode that changes to medium grey. It seems that the background of the entire segment is slightly transparent, and your dark blue in the navigation bar doesn't fit well. If you look really closely at your iOS 13 image you can see a slightly different colour where the segmented control's background alters the blue.
I quickly created a test project and here's one possible solution. You would do this in code inside an #if available(iOS 13.0, *) block so that it only affects iOS 13.
segmentedControl.backgroundColor = .white
segmentedControl.selectedSegmentTintColor = blue
segmentedControl.setTitleTextAttributes([.foregroundColor: UIColor.black], for: .normal)
segmentedControl.setTitleTextAttributes([.foregroundColor: UIColor.white], for: .selected)
This sets the background colour of the entire segment to white to try and provide good contrast against the blue background of the navigation bar. There is some segmented control's own partially transparent view affects this colour, so it appears as light gray, not white. I'm not sure if that can be changed. I think the colour will be very close to the grey below your navigation bar, so could fit well.
The selected segment bubble has the same blue as the navigation bar.
The color of the text then needs changing so that it can be read against the blue or white background: black for .normal (which affects non-selected segments) and white for .selected (the selected segment).
See https://www.icloud.com/iclouddrive/0KGu6wObZOG1_tUIa36DcHbnQ#segment
Depending on how/if you're handling dark mode you might need to use dynamic colours instead of the blue/black/white. This is just a suggestion, of course, but it seems to fit in with your design. If nothing else, it gives you an idea of the kinds of configuration that can be done.
This stack overflow answer says pretty much the same thing and also explains how images in segments are still tinted, but text is not any more: https://stackoverflow.com/questions/56436559/how-to-change-the-colors-of-a-segment-in-a-uisegmentedcontrol-in-ios-13/56874473#56874473
I wouldn't recommend any solution that relies on private implementation of the views inside the segmented control. There's an older answer (before the selectedSegmentTintColor was added) that can almost re-create the iOS 12 style. If you're desperate, you could try that.