How many of today's Internet giants were founded in 1998? Amazon. Plenty of defunct also-rans: Netscape, Ebay, Yahoo, AOL, Geocities, Altavista, Excite, Infoseek, Lycos, Pointcast, Monster.com, RealAudio, Pets.com, Kozmo, AllAdvantage.com, Webvan, DoubleClick. Netflix was in its infancy (~1 year old). Apple was a dying hardware company, a few months from bankruptcy and kept alive by Microsoft's cash infusion.
Google's official founding was several months in the future. PayPal was right around then. Napster was a year in the future. Friendster and LiveJournal were 4 years in the future, MySpace 5 years, Facebook almost 6. SomethingAwful.com, Bodybuilding.com, and Fark.com were a year in the future, 4chan 3 years, Digg 6 years, and Reddit 7. Twitter was 8 years in the future. YouTube was 7. Heroku and Weebly were 8-9 years in the future, which was the same time SquareSpace graduated to more than one employee. Dropbox was 8 years in the future. AirBnb was also 8 years, but didn't catch on for a further 5 years or so. Indeed was 6, Etsy was 7, Indiegogo was 9, Kickstarter was 11, GoFundMe was 11, Pinterest was 12, Patreon was 14.
So to answer your question going by the web's example - probably about 7 years. The "killer app" for the web (communication & coordination, IMHO) started becoming apparent to the mainstream around 2005, with Facebook, Reddit, YouTube, Etsy, Indeed, and Twitter all being founded within +-1 year of that. Interestingly, analogues of all of them existed in 1998 (Classmates.com, DejaNews, RealVideo, EBay, Monster, and...well, nothing really for Twitter), but it took almost 20 years for the underlying infrastructure to get mature enough to get mass consumer adoption.