Date: 201810
Local time of day: 1201-1800
Location: ZZZ.Airport, US
Weather elements / visibility: Turbulence; 10
Aircraft: Citation X (C750)
Operator: Air Taxi
Operating under: Part 135
Flight plan: IFR
Mission: Passenger
Flight phase: Final Approach; Landing
Airspace: Class B ZZZ1
ATC / advisory: Tower ZZZ1
Reporter organization: Air Taxi
Reporter function: Captain; Pilot Not Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Flight Engineer
Human factors: Distraction; Physiological - Other
Anomaly: Deviation - Altitude Overshoot; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Flight Deck / Cabin / Aircraft Event Illness / Injury; Inflight Event / Encounter Weather / Turbulence; Inflight Event / Encounter Unstabilized Approach
Detector: Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued New Clearance; Flight Crew Diverted; Flight Crew Executed Go Around / Missed Approach
Contributing factors: Human Factors; Weather; Procedure
Primary problem: Weather

Narrative: Red CAS message and aural annunciation of 'wind shear' required us to go-around. When we practice this maneuver in the simulator; it is associated with a convective microburst at a lower altitude and survival requires an urgent climb.In this case; there was no convective activity; but moderate continuous chop and gusting winds; both getting worse as we made our approach. I made an immediate PIREP of our lost 20 knots and our go-around. ATC replied 'fly the published miss.' By this time - only seconds had passed - we were well above (at about 2500 feet) the published missed approach altitude of 2000 feet. Noticing our failure to follow his simple instructions; tower gave us a heading; altitude; and a frequency for approach control to re-sequence us for another approach.The second approach; a visual backed up by the ILS; also resulted in a missed approach; because the winds were gusting above our crosswind limitation. Once again; we climbed above the published missed approach altitude. There was no TRAFFIC conflict reported to us.The turbulence on approach was continuous moderate. Everything was shaking so violently that i could not easily read my approach plate; much less the instruments. Even taking hold of the frequency selector knob and changing to the correct numbers was a struggle. I was getting nauseated and my head - even my eyes - hurt.Initially; I meant to chastise my copilot after his first altitude violation. But; I decided not to because this was the correct response in more serious conditions. Judging from his second excessive climb on missed approach; I realized that he did need a reminder from me; which I failed to provide.We subsequently bungled an approach at ZZZ1. Tower issued updated wind report every few minutes. We were close to our crosswind limitation; so every report required another calculation of crosswind. This simple task was complicated by turbulence. On four mile final approach; winds exceeded our limits and we announced our intention to go-around. Tower instructed us to 'fly the published miss'. By then; we had climbed to about 2000 feet; not the required 1500 feet; and were still heading 240; not the required 260. Tower handed us back to approach. We made an uneventful approach and landing at ZZZ2.




Synopsis: C750 Captain reported severe weather resulted in several missed approaches and a diversion.