You're not wrong at all. There are a couple of factors at play, but two good reads are Legacy of Ashes (about the CIA) and The Sword and The Shield (about the KGB)

Essentially there were three things that made the KGB more efficient; history, ethnicity and citizen's rights. Russia, because they were a European power, had a long history of spycraft and intrigue that America, as a new power with (mostly) friendly neighbors, didn't have the same traditions. The USSR was also both the nation and a political party, and had an institutional memory of secrecy and oppression. Combined, this meant that the KGB was a much more seasoned and talented organization. The CIA was formed from the bones of the OSS during WWII, but with a few exceptions we basically started from scratch. If they had google, they would have googled "How To Spy".

Secondly, America is a nation of immigrants and Russia is not. This means that Russia could place Russian-born agents more easily on American soil. Nobody's going to notice a Russian guy in Brighton Beach, but Americans in Moscow were watched much more carefully.

Finally, American citizens are afforded a number of rights (1st and 4th amendment, primarily) that prevents our government from bugging apartments and opening mail without "reasonable" cause. This wasn't the case in Russia, they didn't have to prove you were a spy to bug your apartment, they bugged your apartment to prove you were a spy.

Anyway, the early years of the CIA are fascinating and terrifying. Legacy of Ashes is an amazing read.