"Put the thermometer in the mouth" sounds really weird. In english you would say "put the thermometer in your mouth"
Composites of מַד־ -metre are often written together as the thermometre הַמַּדְחֹם or things like a מַדְגֶּ֫שֶׁם pluviometre.
Is that how they say it in Israel? And not בפה שלך?
Well, I think Hebrew is less obsessed than English to indicate possession, when speaking about body parts. The logic is: Whose mouth do you think you have to put the thermometre in? It is self-explanatory.