The Memory Alpha page on the Prime Directive lists several examples of Picard violating the Prime Directive. Note that for this answer, the Prime Directive will be interpreted in its most absolute form i.e. there being no exceptions to it - pre-warp civilizations must not be contacted.
- 'Pen Pals': Picard declared that the distress message from Drema IV was sufficient grounds for an exception to the Prime Directive. Despite the precautionary measures taken, this still does count as a violation.
- Nemesis: when investigating positronic signatures, the pre-warp civilization of Kolarus III were exposed to Federation technology and Federation personnel, thereby culturally contaminating them.
- 'Who Watches the Watchers': the Enterprise-D was involved in the cultural contamination of the proto-Vulcan society. It wasn't Picard's fault that the holographic projector failed, but subsequent efforts to minimise cultural contamination didn't help matters, so it could be said here that Picard violated an absolute interpretation of the Prime Directive
- 'Homeward': Transportation of the pre-warp society of Boraal II. Again, this wasn't technically Picard's fault, but his decision to resettle the society elsewhere could also be seen as violation of the Prime Directive.
- 'Justice': Picard's interference in the justice system of the Edo society of Rubicun III
So that's five times we see it on screen, but several of these, notably Nemesis and 'Homeward' hadn't even occurred prior to 'Drumhead'.
The article which @Richard refers to lists several other possibilities:
"Angel One": The away team, ultimately Picard's responsibility, gets
pretty close to the line of interfering with the society of Angel I by encouraging its more egalitarian elements.
"Symbiosis": Picard withholds basic assistance from a pair of space
faring races to break a cycle of addiction. His unconventional
inaction could be seen as a form of interference.
"The Hunted": Similar to "Symbiosis", the presence of the Enterprise facilitates certain events (the escape and recapture of
Roga Danar) that Picard later deliberately interrupts to be nominally
"non-interfering" but in actuality to achieve the result he desires
(change in the Angosian government).
"The High Ground": Picard and the Enterprise crew become more entangled than they intended in the Rutian civil war.
"Devil's Due": While "Ardra" was a phony, Picard could be considered to have over-involved himself in the "spiritual" evolution
of the Ventaxians by challenging her authority.
"First Contact": When does a fouled up first contact situation become cultural interference?
"Code of Honor": When your security chief ends up in a fight to the death with the mate of the leader of an alien world, it doesn't
look good on the report.
"Reunion": Arbitrating the future leadership of the Klingon Empire is not exactly Starfleet's idea of a captain's role.
"Transfigurations": An even more incidental effect of the Enterprise going about its normal business, yet still playing a part
in massive societal change.
So, if you were to accept all of these (and quite frankly even I don't accept all of these), it would amount to 14 times on screen we see Picard violating the Prime Directive.
Regarding what is said in 'The Drumhead', of the above cases, 'Pen Pals', 'Who Watches the Watchers', 'Justice', 'Angel One', 'Symbiosis', 'The Hunted', 'The High Ground', 'Devil's Due', 'First Contact', 'Code of Honor', 'Reunion' and 'Transfigurations' are all set prior to 'The Drumhead', making 12 possible violations of the Prime Directive shown on screen prior to 'The Drumhead'. Hence, assuming the only violations of the Prime Directive occurred on screen, then three of those 12 must not be considered violations by Starfleet.