Erdogan, 64, is seeking re-election for a five-year term with hugely increased powers under the new system, which he insists will bring prosperity and stability to Turkey, especially after a failed coup attempt in 2016 that has left the country under a state of emergency. The constitutional changes have allowed parties to form alliances, paving the way for Ince and Aksener's parties to join a small Islamist party in the "Nation Alliance" against Erdogan.
Erdogan remains the favorite to hold on to the presidency - even if he needs a second round on July 8 - but the outcome is likely to be much tighter than he expected when calling the snap polls one-and-a-half years ahead of schedule.
The early parliamentary and presidential elections in Turkey are scheduled to be held on Sunday, June 24, that is, nearly a year and a half earlier than planned.
Erdogan, whose mastery of political rhetoric is acknowledged even by critics, has won a dozen elections but is now fighting against the backdrop of increasing economic woes. Other candidates in the presidential election race are main opposition Republican People's Party's (CHP) Muharrem Ince, newly formed Good (IYI) Party Chairwoman Meral Akşener, the Felicity Party's (SP) Temel Karamollaoğlu, the Patriotic Party's (VP) Doğu Perinçek and the pro-PKK Peoples' Democratic Party's (HDP) Selahattin Demirtaş.
Inflation has zoomed well into double digits - with popular concern over sharp rises in staples like potatoes and onions - while the Turkish lira has lost some 25 percent in value against the USA dollar this year.
Erdogan blamed the coup on his former ally, US-based Muslim cleric Fethullah Gulen, and has waged a sweeping crackdown on the preacher's followers in Turkey. Confident and combative, Ince said "Erdogan you are going!" and called him a "fascist".
"Let this immorality be an example to the world", said Mr Ince, who said he had held 107 rallies around Turkey in the 51 days since his candidacy was announced.
Six candidates are running for president and eight parties have fielded candidates for 600 parliamentary seats. Critics say it will further erode democracy in the North Atlantic Treaty Organisation member state and entrench one-man rule.
Voting started at 0500GMT (8 am local time) and will continue through to 1400 GMT (5 pm local time).
A worker slices meat as a reflection of a portrait of Turkish President Recep Tayyip Erdogan is seen in the window of a Turkish restaurant in Istanbul's Taksim Square on June 20, 2018. A delegation of observers from the OSCE will also be in place.