How do I normalize a database with lots of many-to-many relationships?
It is very clear but you should probably first read some "introduction to relational databases" before you proceed, because from the way you are asking I feel it would be beneficial for you to get some basics first. No offence, it may save you lot of time later.

To give you few fast rules of a thumb


You never ever connect more then two tables with one link. Try to connect tables step by step. Always take a pair of them and try to join them. Do not try to solve the whole thing at once.
you only need the link table if there is so called many to many relationships, i.e. if a client can have many programs and at same time one program can have many clients. It seems to me that it might not be needed in your case.
In all other cases you just place "anothertableID" to one of the tables. In your example it would be good to start with putting ProgramID to ReportTable, ClientTable, PartnerTable. Maybe it is what you need.


To give an more elaborated answer you should give more information about structure of your problem.