Air carrier flight crew flying 737 aircraft reported severe turbulence in cruise flight.

Narrative

At FL350 between ZZZZZ and ZZZZZ1; the CI was 80 and our crossing speed was .80. I had the Speed INTV window open to .78 for the light chop we were experiencing and the seatbelt sign was on. The FO started there was weather about 80 miles ahead of us he was in the 160 mile scale. Looking at radar it was hashed and appeared to be lower than us. I Asked the FO to call ZZZ radio to get weather deviations. We asked for up to 15 miles right of course for weather. Getting closer I changed my range to 40 and it was still hashed. I turned auto off on the weather radar; to see what it painted. It showed a small green area. I then turned auto back on and a lightning bolt symbol appeared on the MFD (Multi-function Flight Display). While getting closer to the weather; it took ATC (no Satcom/CPDLC; just HF's on the aircraft) longer than normal to get us our clearance from ZZZ but we eventually got it and they cleared us to deviate right up to 15 miles of course. We were under optimum altitude and had 25 degrees of bank selected for heading select. I engaged heading select and started my turn to the right. As we leveled from the turn we experienced more than Moderate possible Severe Turbulence. The autopilot disengaged and we lost at least 2-300 feet. I manually took the controls and climbed back to FL350. As I was hand flying I told the FO to make the FA's (Flight Attendants) be seated immediately be seated immediately". He complied. I attempted to turn the autopilot back on and it immediately disengaged again. Then on my second attempt I was able to get the autopilot back on. We continued our deviation around the weather. When I felt it was safe I made the Flight Attendants take your jump seats PA followed by FA's check in. The FA's were all good and no injuries were reported from them or passengers. We finished the flight with no further issues. I wrote up the more than moderate possible server turbulence in the ELB (Electronic Log Book); and on arrival at the gate I talked with dispatcher; local Mx (Maintenance) in ZZZ; and the Chief Pilot.I stated in the ELB write up that it was more than Moderate possibly Severe Turbulence after reading about both in the FOM."