Looking on the internet deeply has found these results:

I am in bed is the most popular phrase on the web.

I am on bed 1156,000 results on the web	I am in bed 12,860,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
I said the patient on bed 12. Which is why... I am putting you on bed rest for the remainder of your pregnancy. I am seeing a doctor on bed for the first time. I have to be on bed rest for hours after the procedure. I got the job because Peanut was knocked up and on bed rest. Don't worry, I am on a bed of roses. I thought you were on bed rest. Okay, while I'm on bed rest, I am going to need your attention 24 hours a day, eight days a week.	It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

I am on bed

1156,000 results on the web

I am in bed

12,860,000 results on the web

More popular!

I said the patient on bed 12.
Which is why... I am putting you on bed rest for the remainder of your pregnancy.
I am seeing a doctor on bed for the first time.
I have to be on bed rest for hours after the procedure.
I got the job because Peanut was knocked up and on bed rest.
Don't worry, I am on a bed of roses.
I thought you were on bed rest.
Okay, while I'm on bed rest, I am going to need your attention 24 hours a day, eight days a week.

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.