I understand light isn’t instantaneous of course, but it’s still fast enough that there shouldn’t be such a noticeable delay.
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Because the signal is relayed through several pieces of equipment. Each time it goes through a relay it has to pause for a small amount of time while the relay looks at what it’s getting and retransmits it. Depending how many legs there are to the trip, it can add a bit of barely noticeable time. Then it happens again on the return trip, making the delay you as the viewer of both ends at the same time see twice as noticeable.
The live transmission between a news anchor and a reporter is typically accomplished via satellite. The news anchor signal is sent to a satellite in space, which might retransmit that signal to one or more other satellites, before it’s sent to the reporter on the ground.
Many of these satellites are in geosynchronous orbit (meaning they stay over the same part of the earth all day long) which requires them to be around 36000 km above the Earth. It takes light 0.12 seconds to travel that distance. But the round trip transmission time is a lot longer, even in the best case scenario:
News station to satellite: 0.12 seconds
Satellite to reporter: 0.12 seconds
Reporter to satellite: 0.12 seconds
Satellite to news station: 0.12 seconds
All of a sudden, you’re at half a second delay in the best case scenario. Then consider the additional delays caused by all the video processing equipment on each side, the additional transmission time if the signal needs to be relayed from one satellite to another (i.e. if the reporter is on the other side of the planet, a single satellite can’t see both sides of earth at the same time), as well as the human reaction time required for the reporter to process the news anchor’s question and formulate a response. You can easily see how the delay can quickly escalate to several seconds.
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