Flight Radar 24 and similar websites connect the dots so you won't see short interruptions. ATC would see your on screen symbol change for every interruption and restoration of Mode C signal. Enroute (long distance) radar updates once every 12 seconds (5.0 RPM) and terminal radar updates once every 4.8 seconds (12.5 RPM).
Mode C has both in and out and is a response signal to a radar ping, hence the name transponder (transmitter/responder) and so Mode C may not be picked up if the antenna is facing away from the radar site. Which is plausible if in a banked turn circling the radar site at lower altitude, the antenna is on the bottom and the bottom is pointed away. Either your mode C doesn't pick up the radar ping or the radar site does not receive a strong enough response, both cases because of being bank toward the radar at the moment of the ping. Especially in mountain regions where alternate radar sites may be blocked. (Modern systems in the USA have multiple radar sites that are electronically combined into a composite for ATC, especially in tracon areas, but mountains are still trouble.)
Mode C is also known more generically as a type of "secondary radar", "primary radar" is purely the microwave ping reflected back to the radar site and [common designs] can only provide accurate 2d position not altitude. So a stealth jet with mode C turned on will show up on secondary radar but not primary, while a GA aircraft with no transponder will show up on primary but not secondary. If radar is totally blocked by terrain your ATC symbol will switch to "coasting" which is just a calculated extension of last known speed and direction.
ADS-B may use the same physical antenna on your plane as the wavelengths are compatible, but it is a totally separate unrelated transmitter receiver system that is not integral with traditional radar. Similarly ATC can see ADS-B info if you are in range of an ADS-B ground site, but again it is distinct from their traditional radar data. As such, even ignoring variations in your transmit strengths, you may be near an ADS-B ground site but far from a radar site, or pointed more at one than the other.