Albert Barnes Commentary Isaiah 38:2

"Then Hezekiah turned his face to the wall, and prayed unto Jehovah," — Isaiah 38:2 (ASV)

Then Hezekiah turned his face toward the wall - This refers to the wall of the room in which he was lying. He was probably lying on a couch next to the wall of his room. Eastern houses usually have such couches or ottomans running along the sides of the room, on which people recline and on which they lie when they are sick.

Hezekiah probably turned his face to the wall so that his emotion and his tears might not be seen by the bystanders, or so that he might better compose himself for devotion. He doubtless wished his prayer to be as secret as possible.

The Chaldee paraphrase renders this as, ‘Turned his face to the wall of the house of the sanctuary’—that is, of the temple—so that it might appear that he prayed toward the temple. Thus Daniel, when in Babylon, is said to have prayed with his windows opened toward Jerusalem (Daniel 6:10). Muslims pray everywhere with their faces turned toward Mecca.

However, there is no evidence in the Hebrew text that Hezekiah prayed in that manner.

The simple idea is that he turned over on his couch toward the wall of his room, doubtless for greater privacy and to hide his deep emotion.