An RDP connection using the MS RDP client would not reveal activities on your local machine to the remote one.
Your description does not specify that you are definitely just connecting directly to a remote machine with a regular RDP client. There are a few scenarios that might affect this answer.
If you connect to RDP through a VPN, this means some or all of your internet traffic might be routed through your company network. The VPN might also set your DNS server. How do you know? In Windows 10, type route print
in a terminal while connected to your RDP remote machine. Is the default route yours? Next try, ipconfig /all
... are the DNS servers yours?
If the RDP client is not the MS RDP client, the provided software would be capable of spying on you, although this is probably unlikely. You could determine the name of the software, and if employee spying was a feature, it would probably be in the marketing materials.
Are you in trouble? Were you looking at things that are legal? Aren't you at home? This might also be a workplace question...