Following the vote, Republic Left announced that three of its members who are also summoned to the Supreme Court on Friday are renouncing their seats in the Catalan parliament. However, she chose to flee the country to live "in exile", as she stated in a letter.
Spanish courts sought Puigdemont's extradition from Brussels a year ago but canceled that petition amid concerns that Belgium might send him back but restrict the crimes with which he could be charged.
Thousands of protesters have filled the streets of Barcelona after Spain's High Court jailed five Catalan politicians who played a role in the region's declaration of independence.
The Spanish Supreme Court charged the former President of Catalonia, Carles Puigdemont, and 12 others, backers of the Spanish region's secession with the crime of rebellion on Friday.
They waved separatist flags and chanted "Freedom for the political prisoners".
Mr Turull is the third candidate proposed by pro-independence politicians since a December election.
The regional parliament now has two months to elect a president and form a government before a new election is triggered.
Rajoy said on Friday he is "not in favour of [Catalonia] having repeat elections".
Earlier this week Catalonia's fugitive ex-president said independence for the region is not the "only option" for resolving the crisis.
Several separatist leaders, including former Catalan vice-president Oriol Junqueras, have been in custody since previous year.More news: Toys R' Us and Babies R' Us close doors
Still, Puigdemont says he fully supports her decision to do so.
It later emerged that she had fled into exile and Spanish media reported that she was in Switzerland.
A judge said the five were serious flight risks. Along with him the Judge passed orders to to arrest without bail, all former regional legislators like Josep Rull, Raul Romeva and Dolors Bassa as well as the ex-president of the parliament, Carme Forcadell.
Judge Pablo Llarena backed his ruling by stating that secessionists had been planning ways to split with Spain for the past six years.
Judge Pablo Llarena decided that the five posed a flight risk, after Marta Rovira became the latest leading pro-independence figure to flee overseas to escape charges. Four of them are in pre-trial detention.
Rebellion charges are punishable with up to 30 years in prison under Spanish laws.
The High court issued global arrest warrants for six more Catalan politicians.
Turull was one of the five politicians who responded to a court summons, only to find that Judge Llarena had chose to send them to jail, denying them bail.
Following the referendum, the central government in Madrid sacked the Catalan regional government, imposed direct rule and called new elections but pro-independence parties returned with a slim majority.
"Exile will be a hard road, but it is the only way I have to recover my political voice", Ms Rovira said in a letter.
Other separatist figures were already in custody or had gone into exile overseas.