For some background on Pre-Columbian contacts across the Bering Strait, you might want to check out this answer from u/TheAlaskan, as well as this answer by u/reedstilt.

I will add that since those answers have been written, there has been new archeological evidence of trade across the Bering Strait in the form of Venetian glass beads found in Alaska that predate 1492.

For why Russia only crossed the Bering Strait in the 18th century, a repost of this answer I wrote:

The major reason is that, despite modern-day Russia being just across the Bering Strait from Alaska, the Chukchi Peninsula (the part of Russia in question) is really, really, really far from where the Russian state was in 1500. Here is a map to give a sense of what Russia (technically, the Grand Duchy of Moscow) looked like in the late 15th century (the green area). The Chukchi Peninsula is some 4,000 miles (as the crow flies) from the eastern parts of the Grand Duchy at the time, and although Muscovy was taking over the fur trade along the White Sea that had previously been controlled by Novgorod, they were still a long way from the Pacific Ocean, and importantly had a number of major states in their path - a group of Turco-Mongol successor states, namely the Khanates of Kazan and Astrakhan on the Volga, and the Khanate of Sibir near what is now Tobolsk in Western Siberia.

Muscovy itself had only just ended its vassalage to the Golden Horde in 1480, and was very busy around 1500 prosecuting the Russo-Lithuanian Wars against Poland-Lithuania, which by 1500 was controlling territory just outside Vyazma and Kaluga, or about 100 miles from Moscow itself.

Muscovite expansion to the east therefore only was able to really begin in the 16th century, after the Western borders were pushed forward at Poland-Lithuania's expense. Astrakhan and Kazan were conquered in the 1550s, and the Khanate of Sibir was slowly subdued in the 1580s and 1590s by Cossacks commanded by Yermak. Once Sibir was subdued, Russian expansion went east at a relatively fast pace, with outposts established on the Pacific in 1639, but even in this period, the Russian population in the vast region was very small, located in outposts, and extremely focused on the fur trade and obtaining iasak, or fur tribute from local peoples. Much like in the Americas, the expansion of Europeans and their trade networks also allowed for the flow of guns, alcohol and diseases to native peoples, with native populations falling drastically, and peoples moving across the Siberian region (some expanding, others contracting).

Russians only reached the Kamchatka region in 1696, and then encountered the Paleosiberian peoples on the Kamchatka Peninsula and the Chukchi Peninsula: the Chukchi, Koryaks, and Itel'men. They were mostly subsistence hunters, with those communities along the coast relying heavily on hunting of sea mammals and salmon fishing. They also resisted Russian expansion more successfully than the violent uprisings of native peoples to the West: a major uprising on the Kamchatka Peninsula had largely driven Russians out in 1706, and the Chukchi north of there caused similar troubles for Russian expansion. The Chukchi lands were not particularly plentiful in the sable furs that Russians craved and that the tsar demanded as tribute, and once hunted wild reindeer herds were depleted, Chukchi were more than willing to take up raiding against Russian outposts as an alternative pastime. The Chukchi even successfully rolled back the Russian empire, defeating a military expedition tasked to "ruin the Chukchi completely", dispatched against them in 1747 and killing its commander, Major Pavlutsky in battle. In 1769 the Russians even abandoned Anadyrsk, which was a fortified outpost established in the 1640s. By this time, in any case, Vitus Bering's expeditions and Russian fur traders had already crossed the sea from Kamchatka to establish outposts in modern-day Alaska.

In short: Russia today is close to North America, but Muscovy in the late 15th century was a much smaller state thousands of miles away from the Bering Strait, and which was busily engaged in land wars with its immediate neighbors. It was only a century after this that it was able to begin expansion eastwards into Siberia, and this was strongly motivated by quick returns for private individuals and for the tsar from the fur trade. Once Russians reached the parts of Siberia closest to Alaska, they hit resistance to their expansion, and ultimately it was easier to go around that into Alaska. The Chukchi would ultimately remain effectively independent of Russian rule until the Soviet period and collectivization. Sadly, they are now mostly known as the butt of stereotypical jokes among Russian speakers.

Source. Yuri Slezkine. Arctic Mirrors: Russia and the Small Peoples of the North