B737 First Officer reported receiving a left IRS drift alert along with increasing alerts and messages on the Captains side resulted in a divert for a visual landing.

Narrative

L IRS DRIFT fms scratch pad message while in cruise. A few min later Captains Primary Flight Display (PFD) amber PITCH mis compare message. Shortly after VERIFY POS Internal Reference System (IRS)-IRS scratchpad message. Control transferred to FO. along with autopilot B. While Captain began troubleshooting; as PF I disconnected the auto throttles. We were in FMC SPEED at M0.76 selected. We were doing almost M.80 and the auto throttles were making zero attempt to fly the commanded speed. Auto throttles remained off for the duration. Capt attempted the call me feature in ACARS but was unable to get Dispatch on the arinc frequencies provided. The next indication was an ALT DISAGREE amber alert on both PFDs. We accomplished the alt disagree non normal checklist. We advised ZZZ that we were no longer rvsm and they gave us a descent. We were almost over ZZZ and there were several thunderstorm buildups around. During the descent the Captains pfd indications continued to worsen. Showing a very steep nose down; increasing angle of bank; with the occasional bank angle alert; altitude split from the FO of about 300 ft; and a slight but growing airspeed split of several knots. We decided to run the Airspeed Unreliable memory items and QRC/QRH. We advised ZZZ of the need to momentarily level off. We confirmed that the FO and standby instruments were accurate. FO flight director was re engaged in heading (HDG) and V/S modes. Per the checklist we re engaged the good autopilot. It immediately started Turning away from the heading bug. FO disconnected autopilot (AP) and hand flew the remainder of the flight. We still did not have an IRS FAULT light overhead. We still were unable to get Dispatch on frequency. We [requested priority handling] with ZZZ Center and advised we wanted to divert to ZZZ1. At this time we suspected a failing IRS; but with the increasing volume of messages; alerts; and the Captains attitude indicator still showing a steep nose down indication; it seemed a new event happened every few minutes. The crew decided to divert to ZZZ since we were in the vicinity; had the field in sight; and the thunderstorms would not affect the arrival; and we knew they could accommodate an intl diversion. We got delay vectors north of the field while we setup for a visual backed up by the RNAV XX. Capt still had no luck with Dispatch/Maintenance Control. He sent a divert acars msg for ZZZ; texted the Flight Attendants (FAs); made a PA. FO hand flew and conducted routine ATC comm. We then finally got the master caution IRS; L IRS FAULT on the overhead panel. Also the Captains PFD attitude information completely went away. We had reviewed the checklist for IRS FAULT and knew it would lead us to selecting BOTH ON R. We were vmc; and setup for a visual approach and busy in the terminal area. We elected not to run that checklist and accomplished a normal flaps 40 landing. The auto brakes did not arm. Taxi to the gate was uneventful.