In short - history. Originally the Piccadilly Line didn't serve Turnham Green at all, there was actually through lines allowing Picc trains to overtake District line ones that had stopped to call.
From the 1930s the present arrangement developed - that Picc trains called in the early morning and late evening but not during the middle of the day. This was to give better connections to the Richmond branch.
Campaigners have long called for more (if not all) Picc line trains to call and I understand this was in the pipeline for when the new Tube stock for the Picc is in service and the signalling/infrastructure upgraded.
Of course now Sadiq Khan has suspended this (due lack of funding in the present climate) who knows when, or indeed if, this will ever happen.
Until December 1932 the Piccadilly line ended at Hammersmith.
The section from Richmond via Gunnersbury to Studland Road junction was opened by LSWR (London and South Western Railway) and later a short (fairly steep) connection was added from Hammersmith to Studland Road for District Railway trains.
The District trains, LSWR trains and Midland coal trains caused the section from Turnham Green junction to Studland Road to be quadrupled early 20th Century, with District using the existing Southern pair of tracks (by then electrified) and LSWR the new northern pair. The District gained a station at Stamford Brook around 1912 and the other 2 (Turnham Green and Ravenscourt Park) had gained island platforms on southern pair of tracks about same time.
Apparently the layout at Turnham Green during the widening towards Chiswick Park was built with diveunders for Piccadilly line trains to be extended to Richmond but it was never used.
Once the LSWR services stopped, the Underground wanted to extend the Piccadilly line over the new tracks, but couldn’t really afford it and nothing happened for about 20 years until 1930. During 1930 Government backed finance became available to relieve unemployment and work started.
Tracks were rearranged from Barons Court so Piccadilly used centre pair, quadrupling was extended to Northfields (where new depot was built). In December 1932 Piccadilly line trains were extended to Northfields and from March 1933 to Hounslow. Initially Piccadilly trains to Hounslow skipped South Ealing and Boston Manor as well as everything between Hammersmith and Acton.
The 1930 scheme assumed Piccadilly line trains would be extended to Richmond, so the diveunders were built to avoid conflicting moves (and explains why the westbound pair have a hump over the Piccadilly trackbed towards Gunnersbury) but this was never done. In later years late night and Sunday Piccadilly trains called at Turnham Green when District frequency was lower to make it easier to change routes.