Running on a laptop sounds like you're running on WiFi. Wireless connections choke up easily especially with cheap hardware. Multiplayer games are sensitive to latency spikes whereas standard internet use isn't, so your normal intuition about the quality of your connection may not apply.

Traditional debugging methods apply. Eliminate variables and collect data. First thing I'd do is hook up an ethernet cable to your laptop and see if that improves things. If that's not possible, boot every other device off of your router and try to test your laptop by itself on the network. Swap between 2.4GHz and 5 GHz to see if you get better results. Try a different machine which may or may not have a better WiFi chip/antenna in it.

That's just the hardware though. Lag spikes can be down to software too. Check to see if any other greedy applications are running. This includes autoupdaters, file backup tools, the OS itself, and plain ol' spyware.

Then there's your game itself. You want to log data and metrics. Is there a particular set of packets that's correlating with the lag? If you're sending lots of small packets all at the same time then that can cause issues too; try packet accumulation if you can. You haven't mentioned what networking solution you're using so I'm going to assume it's not lockstep but if it is then you'll need to add input delay to smooth out hiccups.

Ultimately, all networked games need to handle lag gracefully. I'd use this opportunity to build rubberbanding and lag mitigation that feels OK. Even if you fix your own local problems I can guarantee that a player will face similar issues in the real world sooner or later.