I would like to visit Turkey soon and was wondering if it would be unadvisable to travel to Istanbul during the 10th – 13th of January? Will all restaurants, attractions, etc. be closed for Kurban Bayrami?

definetely not. bayram is a festival as you may already know. but the image/ambiance of kurban bayramı, might be strange and not good for you. don't take into consideration. after first 3 days, everything will become normal.
you're welcomed, beforehand !

Thanks for your help Barkin, much appreciated