So, Catholics don't worship Mary. But there is so much emphasis on Mary that this is the impression that many get who are outside the Church.
In many (most?) Catholic churches there is a prominent statue of Mary (something that the Ten Commandments may condemn? - some Protestants might argue that). I have worshipped at churches in which a resplendent Mary is depicted in a huge painting above the altar, etc.
The most popular prayer in Catholicism would seem to be the 'Hail Mary'. Consider the Rosary where it is said many times more than the 'Our Father'. While the first part of the Hail Mary is found in the gospels, the last part is not.
The major miracle sites that seem to draw Catholics pilgrims today - like Lourdes and Fatima - focus primarily on miracles performed by or through Mary. Among Hispanic Catholics Mary appears even more prominent than Christ. It seems that when someone sees some sort of image, whether on a window or a pizza, it is of Mary. I have visited Ste. Anne de Beaupre several times in Quebec, dedicated to the mother of Mary. The Bible makes no references to St. Ann, of course, and even the Catholic Encyclopedia states that there is some question as to the parents of Mary. Yet, in 1854 (date?) Pius IX declared the Immaculate Conception, that she was conceived without sin. He's the same Pope, by the way, who declared the infallibility of the Pope in faith and morals.
There is a strong suspicion among non-Catholics that Mariology is a cultic side of Catholicism, borrowed from paganism,that there is insufficient evidence in the Bible to merit the veneration (let alone the worship) of Mary. For example, in all of Paul's explicit letters to early Christians, raising many theological points, Mary isn't so much as mentioned once. If she should receive such adoration, wouldn't he have at least hinted at that somewhere in his epistles?
All Christians honor Mary, of course. Some Protestants I know, for example, use only 'Madonna and Child' stamps at Christmas time. Even the Muslims believe in the Virgin Birth, and there actually is much more explicitly about Mary in the Koran than in the Christian scriptures. But the feeling among many non-Catholics is that some Catholics carry it to an extreme, that the focus on Mary is a relic of preChristian faiths that emphasized a female diety, always a Virgin, usually the mother of a Savior-God. There are, of course, miracles associated with the birth of Mohammed, Buddha, and others.
The Catholic Church does such wonderful work, but it really needs to deemphasize some of certain traditions, most of them unBiblical, that increasingly alienate its well-educated parishioners. They want to be faithful but often find that they are called upon to believe the unbelievable, pious stuff inherited from an age before we had telescopes and microscopes, when people believed the world is flat and Jesus literally sat on a throne on the right side of the Father. Acceptable then, but not today.
Keep smiling.