On the fourth hill of the city, to the northwest of the Aqueduct of Valens, was the
church of the Holy Apostles, the most famous church in Constantinople after Saint (Hagia) Sophia. The basilica of the Twelve Apostles was erected in 330 by the Roman Emperor Constantine, who built within the church a large cross-shaped tomb intended for his own burial. He also prepared twelve empty caskets that were to receive the relics of the Twelve Apostles. In 356, the Roman Emperor Constantius brought and deposited under the altar the relics of Apostles Andrew from Achaia and Luke the Evangelist and Timothy from Ephesus. In 550 the church was rebuilt by the Roman Emperor Justinian, designed with the Greek cross plan by the celebrated architects Anthemius of Tralles and Isidorus of Miletus. The church also held the relics of the great Fathers of the Church Saint John Chrysostom and Saint Gregory the Theologian, placed in caskets on either side of the altar. The "Column of Flagellation", to which Jesus Christ had been bound and flogged, was also among the relics of the church. The church's yard was the resting place of the Byzantine Emperors from Constantine and his mother Helen until the 11th century. The graves of all our Patriarchs were also in the yard.
Most of the relics, the gold and silver vessels decorated with precious stones, the icons, the imperial crowns, the magnificent hieratic vestments and other important objects of the church of the Holy Apostles were carried off to Western Europe, when our capital was looted by the Latins in 1204. The Latins plundered the imperial tombs and robbed them of gold and gems. The glorious tombs were completely destroyed in the fall of Constantinople to the Turks (29th May 1453) by fanatical dervishes of sultan Mehmet II. According to the historian Kritoboulos, the dervishes smashed for 14 hours with clubs and steel rods the lyrics. After smashing them, they threw what was left in a lime furnace. In 1461 sultan Mehmet II demolished the church and built a mosque over its foundations,
the Fatih (Conqueror) mosque. It was damaged beyond repair in 1763 by a terrible earthquake and the present mosque was built in its place.
The
Saints Sergius and Bacchus church, a landmark in our ecclesiastical architecture, was built in 527 by the Roman Emperor Justinian, shortly before the Saint (Hagia) Sophia. The church is known to this day as the "little Hagia Sophia", because the general principles of its architecture are comparable with those of our Great Church. The columns are made of coloured marble and the interior of the church shone with its variegated marble walls and the lavishness of its golden mosaic decoration.
In 1509 sultan Bayezid II converted the church into a mosque, the
Huseyin Aga mosque. The Turks destroyed the apse and whitewashed the frescoes and mosaics on the walls.
The
Constantine of Lips monastery is a building complex composed of the church of the Theotokos Panachrantos (the Immaculate Mother of God), the church of Saint John the Baptist and the funerary chapel. The first church was built in 908 to the north of the site by Constantine Lips, a patrician of the Greek Emperor Leo VI the Wise. In the late 13th century the Greek Empress Theodora erected at the center of the complex a new church dedicated to Saint John the Baptist and at the south a small funerary chapel. Excavations unearthed 32 tombs, including those of the Empress Theodora herself and of her daughter Eudocia. The monastery was a true work of art with the alternating tripartite arched windows, the elaborate decoration of toothed bands and meanders and the interplay of red and white brick blocks.
In 1453 sultan Mehmet II converted the monastery into a mosque, the
Fenari Isa mosque. The north part of the peristyle of Saint John was destroyed. The mosque suffered severe damages by fire in 1622 and 1917. After the first fire the Turks destroyed the columns. After 1917 the monastery remains in ruins.
The
Myrelaion (Holy Anointing Oil) monastery[B/], was built in 920 by the Greek Emperor Romanus I Lecapenus. Several members of the Macedonian and Comneni dynasties were buried in its basement crypt. The architecture of the chapel, the only surviving part of the monastery, is considered a masterpiece.
In 1203 the Latins burned the monastery. In 1574 sultan Murat III converted the monastery into a mosque, the Bodrum mosque. The mosque was destroyed by fire twice in 1784 and 1911. A fine portrait of a Byzantine princess revealed during excavations is now missing.
The [B]Jesus Christ Pantocrator monastery was built on a hill, to the right of the large avenue leading from the Golden Horn to the Aqueduct of Valens, in 1136 by the Greek Emperor John II Comnenos and designed by the architect Nikephoros. This monumental complex is the greatest church built in Constantinople after the time of Justinian I. The Greek Emperor Manuel I Comnenus brought from Ephesus the marble "Stone of Unction", on which Christ's body had been anointed before the entombment. Large crowds gathered in the monastery every day from all corners of the Greek Empire to venerate this sacred relic. In the grave chapel a number of Greek Emperors were buried including John II, Manuel I and Manuel II Palaiologos.
During the Latin occupation (1204-1261) the monastery was looted by the Venetians. The icons, sacred vessels and holy relics of the Pantocrator still shamelessly decorate the church of Saint Marco at Venice. In 1453 sultan Mehmet II converted the monastery into a mosque,
the Zeyrek mosque. A huge emperial sarcophagus from green specked stone with crosses on the four sides is used as a foot-bath (!!) by the Turks entering the mosque. The rough-and-ready appearance of the church makes it seem as if the conquest of Byzantium has just happened, as if the ghosts of the monks cannot be far away.
The small but elegant
Saint John Baptist monastery in Trullo was probably built in the 12th century. In 1520 pascha Ahmet converted the monastery into a mosque.
On the fifth hill of the city, overlooking the Golden Horn, is the
Theotocos Pammakaristos monastery, built in 1305 by Michael Tarchaniotes Glabas, one of the best and finest works from the late Byzantine era. Many travellers and scholars who visited Constantinople described with admiration the monastery of Pammakaristos. Though severely damaged, the surviving mosaics reflect the brilliance and high quality, the remarkable style and technique, the classicizing trends, and in general the culture and spirit of the Palaeologan revival. The funerary chapel, consecrated to Jesus Christ, with its elegant proportions, the fine walling with the ornamental toothed bands, the slender shallow niches and wide blind arches pierced by double and triple lights and the two elegant domes, is one of the most important examples of ecclesiastical architecture in the 13th century.
In 1587 sultan Murat III converted the monastery into a mosque, the
Fethiye (Victory) mosque. In the main chapel the Turks destroyed the apse to the east. Most of the unique and holy mosaics were whitewashed. In the funerary chapel the portraits of Tarchaniotes and his wife were destroyed.
Link:
http://www.e-grammes.gr/2000/06/church_en.htm Wanted to share the information even though I think the passage is too biased.

The monuments with Turkish origin can't be preserved well, either. Don't be surprised if you see 400-year old baths turned into shops, if you see great mosques with broken doors, etc.

It will take time telling many people that preserving monuments is crucial.
Regards...
P.S. I searched for a similar topic, but couldn't find one.