You probably have the same issue I have. The longer the laps the higher the chance of me disconnecting after completing one lap, happens mostly in races, but can and will happen during Practice or Qualifying too.
That issue is down to your internet provider and I myself drive many of the races with 2+min laptimes on mobile data through USB-Tethering to avoid those disconnects.
The description of your problem is 1:1 the same for me in terms of the live timing etc. so if you can you'll need to drive with mobile data instead, more on that here: http://www.koelschbierbude.de/component/
In case the mobile data usage I pointed out in that post is too high for your liking you can easily turn down your connection quality in the lobby from LAN (best) to ISDN and you'll most likely use less than 100MB during an evening of racing. I don't recommend using either of the two Modem options since that will lead to you not seeing all cars.