So in a nutshell i am a streamer (small) who wanted to start a new game stream "series" being GTA V while playing the game during streaming it runs amazingly well but as soon as you check the stream it is super laggy and jumps like huge amounts of seconds then freezes on that frame then hops to a whole new one then repeats
This is not due to the game, as you said the game runs fine for you.
This may be because you are trying to stream with an upload speed that is too slow to stream.
Firstly, what resolution and bit rate are you trying to livestream at? To check this, go into the settings of the software you are using to livestream. You may be livestreaming a video resolution that is too high for your upload speed.
Secondly, what is your upload speed? Go to speedtest.net and do a speedtest, and report the "upload speed" to me. The minimum upload speed for livestreaming at 720p is 3mbps (megabits per second), 1080p is 6mbps, and 4K is 32mbps. Remember that this is the MINIMUM upload speed, you need more than the minimum requirement to not strangle your internet connection.
Another possible reason is that your PC is not powerful enough to run a game AND livestream it simultaneously, but this is unlikely.