"...the shroud was...exposed to a fire...in the 1300s..."
It is true that the shroud was exposed to high temperature in a fire, as you state. Russian scientists took a piece of cloth similar to that of the shroud, of 1st Century vintage, carbon dated it, exposed it to high temperature similar to what the shroud experienced, carbon dated it again.
The first dating by the Russians confirmed 1st Century, and after the high temp exposure the dating showed it to be much younger, like of recent vintage. With this data, we cannot say that the carbon dating of the shroud showing 13th Century proves it a fake. (Actually, I believe the dating shows the shroud to be 14th Century.)
The shroud has a history that predates the 13th or 14th Century. Early images of Jesus have been found that are similar to the image of the shroud, and appear to be copies of that image on the shroud. It has been determined that the image is not painted, nor inked onto the fabric. It is a photographic negative. It definitely appears to be the image of a crucified man who was scourged without mercy. The image shows that the man wore a crown of thorns (not a circular "crown" but a crown like a skull cap, pressed hard into his head. Residues on the shroud have been identified as human blood. The marks on his back indicate that he carried a beam, the cross-piece of a cross.
Fake? I do not know about the authenticity of the shroud. Is there enough evidence to claim with certainty that it is a fake? No, I do not believe that there is. I once thought so, but not now.
I do know one thing for sure - the man, Jesus, the Christ, the Son of the Living God, is real. He was here on this earth, born in the flesh, he died on the cross nearly 2,000 years ago, and he rose and lives. He died for me. He died for you. He died that we all might live. The life He gives is there, freely given, we only have to accept that gift and it becomes ours.
This is false. No amount of heating can change the isotopic content of the Carbon in anything. Even allowing for the collection of soot in the fibers would not skew the results of a first century cloth enough to give a date in the 13th Century. The reason for the erroneous C14 tests is now very well known and can, and has been, proved. The sample taken from the Shroud for the C14 tests was cut from an area that had been invisibly rewoven in the 16th Century with new, distinctly different threads.
FYI: The original C14 tests by the Arizona, Zurich, and Oxford labs reported creation dates of 1260 - 1390, a time spanning both 13th and 14th centuries.