Looks like Elizabeth Mary HARTY was named Lizzie at birth.

"Ohio, County Births, 1841-2003"
Name: Lizzie Harty
Event Type: Birth
Event Date: 08 Jul 1882
Event Place: Youngstown, Mahoning, Ohio
Registration Date:
Registration Place:
Gender: Female
Father's Name: Michael Harty
Father's Birthplace: (Tipperary, Ireland according to death reg)
Mother's Name: Bridget Barnnett (Bennett on death reg)
Mother's Birthplace: (County Kerry, Ireland according to death reg)
Reference ID: v 3 p 111
GS Film number: 906502
Digital Folder Number: 004017174
Image Number: 00381

Looks like Elizabeth and the rset of her siblings are living with their elder brother Matthew

1900 United States Federal Census
Name: Elizabeth Harty
[Elizabeth Hartz]
Age: 17
Birth Date: Jul 1882
Birthplace: Ohio
Home in 1900: Youngstown Ward 6, Mahoning, Ohio
Race: White
Gender: Female
Relation to Head of House: Sister
Marital Status: Single
Father's Birthplace: Ireland
Mother's Birthplace: Ireland

Matthew Harty 26
Helen Harty 25
Harriet Harty 21
Elizabeth Harty 17
Juliaetta Harty 16
Mary B Harty 14

This looks like the family in 1880

1880 United States Federal Census about Mattie E. Hartz
Name: Mattie E. Hartz [Mattie E. Harty]
Age: 6
Birth Year: abt 1874
Birthplace: Ohio
Home in 1880: Youngstown, Mahoning, Ohio
Race: White
Gender: Female
Relation to Head of House: Daughter
Marital Status: Single
Father's Name: Micheal Hartz
Father's Birthplace: Ireland
Mother's Name: Bridgett Hartz
Mother's Birthplace: Ireland

Micheal Hartz 35
Bridgett Hartz 34
Phillip J. Hartz 7
Mattie E. Hartz 6
Ellen M. Hartz 5
Mary T. Hartz 3
Ida Hartz 2
Harriet Hartz 1

Ohio, Births and Christenings Index, 1800-1962
Name: Matthew Edward Harty
Gender: Male
Race: White
Birth Place: Youngstown, Mahoning, Ohio
Birth Date: 27 Jan 1874
Father's Name: Michael Hartly
Mother's Name: Bridget Burnett (yet a different surname)
FHL Film Number: 906501

"Ohio, Deaths and Burials, 1854-1997"
Name: Michael Harty
Gender: Male
Death Date: 06 Oct 1894
Death Place: Youngstown, Mahoning, Ohio
Age: 49
Birth Date: 1845
Birthplace: Ireland
Occupation: Saloon Keeper
Race: White
Marital Status: Married
Indexing Project (Batch) Number: B07110-4
System Origin: Ohio-EASy
GS Film number: 906672
Reference ID: v 4 p 33

This looks like your Michael and Bridget ? but they must have entered Canada before going to the USA

"Ontario Marriages, 1869-1927"
Name: Michael Harty
Event Type: Marriage
Event Date: 20 May 1872
Event Place: Douglas, Renfrew, Ontario, Canada
Age: 28
Birth Year (Estimated): 1844
Father's Name: Philip Harty
Mother's Name: Ellen Bonfield
Spouse's Name: Bridget Bennet
Spouse's Age: 27
Spouse's Birth Year (Estimated): 1845
Spouse's Father's Name: Mathew Bennet
Spouse's Mother's Name: Johanna Lyons
Page:
GS Film Number: 1862698
Digital Folder Number: 004529103
Image Number: 00675

Go to following site:

https://familysearch.org/search

Enter Michael Harty and under spouse Bridget search and you will find most of the childrens christenings, some marriages & deaths.

Make sure you only look at only the Ohio ehteries.