It has been determined in the medical profession that it is not necessarily how high the fever goes in a child but how fast it goes high that is off issue. A very fast rise can cause a seizure. Febrile seizures are considered benign. In the old days a kid who got one might be on anti-seizure meds for a long time, Not so anymore.
Young children will often got up to 104 or 105. If the fever rises slowly no seizure will occur.. It it shoots up fast a seizure may occur and usually this will happen before you even know you are heading for a problem' Of course there are exceptions to every rule but generally this is how it is... and they dont run around the pediatric units anymore trying to prevent seizures that were not going to happen anyway, like they use to
Your daughter is in that inbetween stage being 13, no longer a child but not yet an adult.
Give tylenol or Ibuprofen only if she is pretty uncomfortable, where the fever is giving her distress of headache, pain, lethargy, And you do not need to give so much or repeatedly to bring it to normal.
A fever is good and it a reflection that the immue system is working as it should. At that age I would not treat the fever most of the time till it hit 102 or 103, but still based on how she feels.
As far as a thermometer goes, I can by without one.. If you are in the fear mode it can push you to treat where maybe you should not. But I used them and held my breath and practiced restraint.. only give meds if they are pretty uncomfortable and certainly if they start loosing the desire to drink.
If she is getting glassy eyed, delerious or quite red faced give tylenol. apply cool compresses to the forhead.
You do need to know if it is a bacterial issue or a viral issue. Bacterial issues need further treatment with a physicians assistance generally.
Brain damage is pretty dang rare from a fever, it would have to be over 106 for a time. 106 still is not unsusal for a small child but I certainly would be working to bring it down some with tylenol or Ibuprofen. If tylenol does not work give Ibuprofen or alternate them a few times at least 4 hours apart. Again, cool compresses.
My kids would be quite active at 104 and not want to go to bed. At 105 was crash time for them and I would give something... but I always used homeoapthics for the most part which takes some extended skill to use. We rarely used anything else.
A rectal temp is the most accurate. If you take it under the arm add one degree. Oral is somewhere inbetween and is generally close enough.