The Iceland lorry ended up in the grounds of St John's Church following the crash on Woodfield Street, Morriston, at about 04:50 BST on Wednesday.
A spokeswoman for Iceland said the driver was taken ill at the wheel.
He was taken to Morriston Hospital "as a precaution", South Wales Police said.
Officers are investigating the cause of the crash, it added.

TV is the leading medium and three major broadcasters - state-owned TVP and private TVN and Polsat - dominate the market.
The main players in radio are Germany's RMF Group, France's Eurozet and state-owned Polish Radio.
There are more than 300 newspapers, most of them local or regional. The most popular paper, the Fakt tabloid, is owned  by a German-Swiss group.
The constitution guarantees freedom of expression and forbids censorship.
However, the newly-elected conservative Law and Justice government introduced measures in late 2015 to allow ministers to appoint the heads of TVP and Polish Radio, prompting an unprecedented European Commission inquiry into any potential threat to freedom of expression.
There were 28 million internet users by 2016 (Internetlivestats.com). Facebook is the leading social network; there were more than 20 million Polish accounts by 2015.

The anti-IS group Raqqa is Being Slaughtered Silently said a water pumping station had been hit, along with the Taj Hall and Furousiya area.
IS-linked news agency Amaq  also said Raqqa's water supply had been cut.
Russia said its bombers had targeted an arms depot, a chemical weapons factory and a training camp.
A defence ministry statement said the strikes had inflicted "significant damage" and that a large number of militants had been killed.
Raqqa is Being Slaughtered Silently (RBSS), whose activists have reported on developments on the ground in Raqqa since IS militants seized control of the city in early 2014, said the water pumping station hit in Thursday's strikes was located in the nearby village of Kasrat.
Residents found their water had been cut off "totally" afterwards, it added.
Amaq also reported the "interruption of the water supply in all neighbourhoods".
RBSS said 20 civilians had been killed and 50 others wounded in the air strikes, but the UK-based Syrian Observatory for Human Rights put the death toll higher.
It said 24 civilians had died, along with six others whose identities could not yet be confirmed.
Raqqa, estimated to have a population of between 250,000 and 500,000, has become the de facto capital of the "caliphate" whose creation was proclaimed by IS two years ago after it took control of large swathes of territory in Syria and Iraq.
IS militants are currently under pressure from two separate offensives west and north-west of Raqqa by Russian-backed Syrian government forces and an alliance of Kurdish and Arab fighters supported by the US.

Shocked onlookers swam across the boating lake at Stanborough Park, Welwyn Garden City, on Sunday after the car became submerged in the water.
The pair were put in a boat and taken to land. The man, who is from Potters Bar, was taken to hospital but discharged later in the day.
Police said he was "shaken" but "not injured".
As reported in the Welwyn Hatfield Times, video from Alex Kerr on YouTube shows three people in the lake, trying to smash a window of the Nissan Qashqai.
A spokeswoman for Hertfordshire Police said they eventually helped the man and his dog out of the car via other means.
She said the force was called at about 12:45 BST and the car was pulled from the water by the fire service shortly after 16:00.
A delivery truck crashed into parked cars before hitting a church wall in Swansea.
[NEXT_CONCEPT]
Poland's broadcasting market is the largest in Eastern and Central Europe.
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Air strikes on so-called Islamic State's Syrian stronghold of Raqqa have cut the city's water supply, with 20 civilians reported dead.
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A man aged in his 70s and his dog were pulled from a car that had plunged into a lake.