...the browser seems to be trying to download them in order to play them rather than stream them.
To echo what @Offbeatmammal said in the comments, if you're using MP4 files, you need to ensure the MOOV atom is at the beginning of the file. Without it, the browser doesn't know what byte offsets to request.
Ideally, encode your video files as fragmented. In FFmpeg:
ffmpeg -i ... -f mp4 -movflags frag_keyframe+empty_moov output.mp4
See also: https://stackoverflow.com/a/9734251/362536
That should allow the client to stream the MP4 files from any web server that supports HTTP/1.1 range requests. (Most all do, unless configured otherwise.)
However, there is another point to address:
The problem is some of the video files are 1Gb or larger and the connection to some clients is rather slow...
While fixing the streaming issue means the clients won't have to download the whole file first, they still need the bandwidth to keep up with the stream. If it's possible they won't, you'll want to implement some sort of transcoder.
I would recommend using an existing segmented streaming method such as DASH or HLS. HLS is currently the most compatible, thanks to Apple's platform policies. Either will enable adaptive bitrate switching, which will allow slow clients to automatically switch to a lower bitrate stream that they can smoothly keep up with. That way, slower clients can still see the video, albeit a lower quality one, while fast clients can get the full quality video.
You can use FFmpeg to do the transcoding and HLS playlist creation.
I'm not sure if PLEX or something like that is able to do it dynamically as there are hundreds of videos and new videos added all the time.
As for when you do this transcode, I suppose it depends on how much load you're looking at. If this is just one or two people viewing the file, you can transcode on demand if your servers can keep up. Ideally, you have at least a couple stream variants around for less popular files, and add more later if needed.
If you're doing this live, I'd recommend doing all of your transcoding up front. You can always prune old files/variants if you need the storage back.