Supernatural pregnancy: The baby is conceived normally, but during pregnancy something mystical happens to the mother and/or the fetus. Unfortunate women subjected to experimentation either through a Mad Scientist or a Back-Alley Doctor can end up with this.
Both. The baby is conceived and carried to term through supernatural means.
This trope is a good way to give rise to empowered children; depending on the pregnancy and the supernatural means involved a mother could also birth a Fetus Terrible. However, there's also a large possibility that the phlebotinum involved could kill the mother, as well. Note, however, that test tube babies and other methods not involving natural conception are not examples of this, although some tropes could overlap - such as Designer Babies, which usually involve more "science" and less "magic."
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Horrific example from Berserk: Guts and Casca's child, conceived in their moment of love by the waterfall, was corrupted by the demonic seed of the newly-born Femto, who raped Casca during the events of the Eclipse.
From the same series, Ganishka's demon-soldiers are created by dropping heavily-pregnant women into vats of demonically-charged amniotic fluid which instantly corrupts and mutates the foetus into a demon, who then claws and bites his way out of the womb.
In the 3rd OVA to Fushigi Yuugi, Miaka and Tamahome conceive a baby on their wedding night. Three months later, Mayo finds the Book of the Four Gods again, and the pregnancy is supernaturally transferred to Mayo (while Miaka goes into a coma), because Mayo has such strong feelings for Tamahome. (She takes full advantage of the pregnancy and her status as Priestess, much to Tamahome's dismay.) Even though it seems all is lost, it's actually a Xanatos Gambit on the part of Miaka, in order to save the Universe of the Four Gods. After the fetus is used as a Living Macguffin in a Metaphysical Place to summon Suzaku, the baby is transferred back into Miaka, who wakes up and the pregnancy continues normally. (Mayo, too, goes on to live a normal life.)
The Invisible Woman of the Fantastic Four had already had a difficult pregnancy with her first child, Franklin. Susan's second child was conceived in the Negative Zone, and resulted in a stillbirth.
The plot of the comic book series America's Got Powers begins when an alien crystal falls to Earth and gives off a burst of energy. All of the pregnant women within range, regardless of how far along they were, immediately give birth to fully-developed, healthy, super-powered babies.
The Boys: Billy's wife dies when her superpowered fetus (she'd been raped by a super) explodes out of her uterus, floating in the air and firing Eye Beams. He kills it with a table lamp.
In the My Little Pony franchise, the Lady Land nature of the series doesn't stop Shipping, and coming up with magical means for two mares to have foals often comes up.
Rosemarys Baby involved a young woman getting prescribed mysterious drugs by her doctor during her pregnancy, which she comes to suspect aren't legitimate treatments but something more sinister... but it is too late, and she births an Enfant Terrible (implied as possibly The Antichrist).
In The Phantom Menace, Shmi Skywalker says that Anakin has no father, which Qui-Gon assumes to mean that he was conceived by the will of the Force. In Revenge of the Sith, Palpatine's discussion about Darth Plagueis include a pointed comment that suggests Plagueis's experiments may have been responsible for Shmi's pregnancy.
Blade's mother was bitten by a vampire not long before he's born. The result: He's a Dhampyr.
Tangled: The pregnant queen becomes deathly ill, and the kingdom's soldiers seek out a mystical flower that could heal her. When she ingests it, she is cured, and her daughter is born with hair with healing powers.
In Angel this happens to Cordelia twice and Darla once.
In an early episode Cordelia has a one-night-stand with a guy who turns out to be a human 'donor' for the sperm of a demon he worships. It ensures the human is extremely lucky with women and that any woman he sleeps with becomes immediately pregnant with the demon's offspring, carrying the offspring to term in only a few days and dying in the process of giving birth. Cordelia is saved just before she gives birth.
Cordelia was used by a Power That Was as an incubator for that Power's own birth after that Power had instigated several miracles which were pre-requisite for It's own birth.
Vampires can't have children, but Angel and Darla did, after having both been brought back from Hell. The baby was human with certain, unexplained demonic connections. Because her body was dead during the pregnancy, Darla was unable to give birth so had to stake herself to save the baby. The pregnancy and birth was also one of the miracles required by the Power That Was for It to be born as it required this child, Connor, to sire it into being via Cordelia's body.
Tim doesn't believe that Corinne's baby is his because she's showing after only a few weeks of pregnancy. She insists that it's Tim's baby, and ends up giving birth after only a month or so. It turns out that the baby is possessed by the devil.
Mary is pregnant with Burt's child, but she's not sure it's his - she conceived at around the same time when he had been kidnapped by aliens & replaced by an alien doppelganger, so it might be half alien.
On Once Upon a Time, Cinderella was pregnant when the Dark Curse brought her to Storybrooke, where because of being essentially frozen in time, she was pregnant for 28 years. Delivery was also complicated because she had tried to renege on a deal with Rumplestiltskin.
In season 9 of Stargate SG-1, Vala is impregnated by the Ori (god-like aliens) with their version of the Messiah. For a while, Vala just knows that she didn't have sex but she isn't sure how it happened. It isn't until late in the pregnancy that she finds out the child is "the will of the Ori."
In Twilight: A character note Bella ends up impregnated by a vampire note Edward and giving birth to a Half-Human Hybriddhampyr that ends up growing supernaturally quickly.
In the Elder Races stories by Thea Harrison, the pregnancy resulting from the marriage of Dragos Cuelebre and Pia Giovanni was complicated to say the least. For starters, Dragos is actually an immortal dragon in human form and Pia is the offspring of a human and a unicorn in human form; Pia herself can turn into a unicorn.. Since the pregnancy was utterly unprecedented, no one knew for sure how long it would go on or what form the offspring would take. Fortunately at nine months, the fetus transformed itself in utero to a human baby and was born normally
A Song of Ice and Fire: Daenerys's pregnancy starts off normal, but after she asks a witch to perform Blood Magic to save her husband, the witch complicates Daenerys's pregnancy, leading her to birth a horrific reptilian baby that may have damaged Daenerys's womb and rendered her incapable of bearing children forever. (Or did it?)
Thief of Time: the Anthropomorphic Personification of Time goes through one of these, essentially having the same birth twice, with the end result being one soul going in two children. Nanny Ogg, who was the midwife, was quite surprised, as she told herself (literally).
In Dune, Lady Jessica drinks and transforms the Water of Life for the Fremen seitch while pregnant with Alia. Due to that Alia is born with uncanny psychic powers and the memories (and as seen later, multiple personalities) of all her ancestors—a condition the Bene Geserit foresaw (or probably had seen from prior experience) as coming from such circumstances, to be avoided as "abomination".
In The Ramayana, the god Vishnu turns himself into celestial porridge (no, seriously) that King Dasaratha is supposed to feed to his wives. All three of them subsequently give birth to sons that are endowed to different degrees with Vishnu's essence.
In the Forgotten RealmsPools of Radiance trilogy, Kern's mother was affected by magic during her pregnancy. Kern gestated at an amazing rate, but didn't age any faster. He's also nearly immune to magic, including healing magic and other helpful magic, which appears to be a result of what happened to his mother.