1) Check your Internet Contract. It isn't how 'fast' you believe your playing (i.e. I have 20Mpbs Internet) it is how fast your UPLOADING which is usually less than HALF your download (i.e. 20Mbps DOWN, 3Mbps UP).
2) ANY of the TWITCH regulars post the SAME thing over and over and over. So let me break it down.
A) ANY streaming KILLS 50% of your FPS. If your playing 60FPS expect to perform 30FPS when streaming.
B) ANYTHING BELOW i7 will kill that FPS number for streaming because your asking your CPU to do a bunch of processing just for the GAME (see your % usage of your CPU at the most INTENSE gaming point, best to monitor with MSI Afterburner then enable the OSD - on screen display - of each thing to monitor IN GAME as you play so you have real AS I PLAY measurements) then you want to add on recording both the video and audio rendering, converting to the stream type (aka MP4, MPEG, AVI, FLV, etc.) and COMPRESS that audio and video into one streaming file takes a HUGE amount of processing. So a multicore (6-8) is best to perform both separate things going on (mind you that means no streaming music, no Skype, no email checking, etc. minimized or in tray).
C) HIGH END VIDEO CARDS, which are expensive ALSO NEED HIGH END COOLING AND POWER. Just because you want to get a 970 doesn't mean you have the POWER much less the AFTER MARKET COOLING to make it work JUST to game, much less to stream as well.
D) Make sure your NOT SHARING YOUR INTERNET WITH FAMILY / FRIENDS! If you want to Twitch/otherwise you need a dedicated Internet connection just for yourself or expect issues with gggglittchhhsssss and so on when streaming. Is why so many play 'at night' when everyone else is in bed so they aren't watching NETFLIX, Youtube, etc. causing impact on your 'performance' and 'stream'.
3) Your '120FPS' is a bogus setting and I bet confusing it with your '120Hz' display. FPS does not mean display HZ performance, and display HZ performance does NOT mean FPS. They are two different things. If your running a normal LCD screen it has a 60Hz display, whether your pumping 3FPS or 300FPS the display will show 60HZ no matter what, how it does that (duplicating the 3Frames persecond to match the 60HZ OR dropping 240Frames to match the 60Hz) is what Vertical Synch helps stablize so you don't have tearing / stutters. IF you have a 120Hz screen (like I do) then again it BEST to match the FPS to it (trying to get 120FPS) but it isn't essential, as long as your running over 50FPS your game performance will be smooth and good.
4) Streaming is NOT in FPS, but in rated video signals, i.e. 480p/i, 720p/i, 1080p/i. The HIGHER the video signal output the MORE demand is need to perform that much information for that HIGHER level detail (not even considering sound quality performance, compression or enhancements which is additional demands). IF you spent the monthly high cost (like me) for the highest Internet (my cast Time Warner gives me 300Mbs Down 20Mbs UP), invest in a i7 / 960-970, and make sure your QoS (Quality of Service) in your router is putting your stream at the top of the demand list (that mean's Mommy going to get very upset when her Sons of Anarchy episode..is... gl..iittchhing.. sttutttering.. because your hogging the bandwidth), then pumping out 1080P is possible most of the time. Normally most will look to do 720p streams to as low (like on your system now) at 480p to be 'smooth' showing on the people viewing.