I have now found your John Turner on a passenger list to Oz with Jane Atkins, a 17 year old bonnet maker, on the same ship.
http://www.slsa.sa.gov.au/fh/passengerlists/1838Eden.htmThey were on The Eden out of Portsmouth arriving SA 24th June 1838. Consecutive passenger numbers suggest John, a brick-maker, was with 2 of his brothers, Henry James Browning aged 21 (born c1817) & William aged 17 (born c1821). John's age on that passenger list is 20, giving him a birth year of c1818. The only member of the family with the middle name Browning listed against his name however, was John's elder brother Henry who was a Farmer.
John and William were married in a double wedding (Adelaide 20th Aug 1838) both having elder brother Henry as their witness.
William's new wife was 21 year old Amelia Goble who was also on The Eden in 1838 with her younger brother Charles.
It may help narrow the search down for parents if all 3 brothers can be found being baptised in the same West Sussex Parish. Alternatively, assuming both Henry and William outlived their brother John, you may find one of them has put the names of their mutual parents on other Australian documents. Look up Henry James Browning Turner who died 14 Feb 1893 in Mallala, South Australia for example. An unexpected bonus is that Henry and his family are well recorded and there is a photo of his gravestone on the Internet.
http://www.australiancemeteries.com/sa/mallala/mallala_hsimages/turner_henry_elizabeth.jpgRoy G