As @cayars said, this is a file format/codec issue and doesn’t have anything to do with the web browser at all. If your media were in the format he suggested above, it likely would play on EVERY device you can use Plex clients on, including the Web app. The format @cayars suggested plays on almost every other Plex client app they make without transcoding in most cases.
Because it’s not in that format you are having issues with your media on that player. And will continue to have those issues as long as you don’t convert your media.
But, you know what, don’t take either his or my word for it. Run a test yourself. It’s not hard to do. Download Handbrake, take a single file that you know has problems, and convert it to MP4, H264, with high profile and an AAC stereo audio track.
Put it into it’s own, completely separate folder not already in a library on your PMS machine and then point a library at that folder and let Plex find it. Then try to play the converted file. I would bet you a cup of coffee or maybe even a beer that it works after this conversion.
And if it works, that tells you everything you need to know to fix your issues with all of the rest of your media.
IMO, Plex has placed entirely too much emphasis on transcoding and not enough on Direct Playing the media. Plex can convert the media to something that almost anything the clients can play, but this fixes it in people’s minds that they don’t have to worry about how they store their media. They just have to have it. In reality nothing could be further from the truth.
Preparing the media for Direct Play is as important as setting up PMS on a box that can handle the odd transcode, or having enough bandwidth to stream it or having the media your users want to view. In fact, IMO, it’s MORE important than all of those things. I’ve streamed to 5 local users simultaneously on a device that didn’t have enough power to transcode to even one single client. Because I prepped the media before I handed it to my PMS to stream.
It’s your call. Keep having the problems you are having, or run a test to see how it works. Start using your PMS smarter and not making it work harder. I know… You want to watch your media, you don’t want to have to worry about how this all works. So after you run this test, look into @cayars scripts and give them a whirl.