The key clue is that people still hear your game and music when the mic is muted on the headset. That means the sound isn't leaking through the mic at all — your PC is sending its own output as your "mic". Check these:
1. Discord → User Settings → Voice & Video → Input Device: pick the actual mic (often "Microphone (Realtek…)"), not "Stereo Mix", "What U Hear" or "Default".
2. Windows: right-click the speaker icon → Sound settings → More sound settings → Recording tab. Right-click "Stereo Mix" → Disable. Then set your mic as Default Device.
3. Still in Recording, open your mic → Properties → Listen tab: untick "Listen to this device".
4. The splitter must be the right kind: one 4-pole (TRRS) female socket to two plugs — green into the headphone jack, pink into the mic jack. A plain stereo splitter (two headphone outputs) won't work. If your PC has a single combo jack (laptop or front panel), skip the splitter and plug the HS35 straight in.
5. If you use the front panel, try the rear jacks on the motherboard. Front-panel wiring sometimes mixes audio into the mic.
You can check the mic alone at
https://mictests.com/: play some music, mute the headset mic, and the level should drop to zero.