I cannot find a way to see my laptimes over a stint in single player, SURELY there is a way, that's what practice is for :/
You can see your lap times in the setup screen. For the big timing screen, you should do a practice session in a race weekend.
I can't do without this beautiful little piece of freeware https://forum.sector3studios.com/index.php?threads/secondmonitor-timing-status-app.9587/