Except for a period between 2009 and 2016, Israel and Turkey always had diplomatic relations. It is impossible to give trump credit for original diplomatic relations being setup in 1950 (when he was 4 years old), or being reestablished when Obama was President. In 2018, ambassadors were expelled again, but embassies remained. That is the only thing that happened while trump was in office, and it was not good.
Turkey recognized Israel in 1949, and first exchanged legations in 1950. The legation was downgraded to a "Charge d’Affaires" in 1956, but normal diplomatic relations continued. In 1963, it was upgraded to a legation again. In 1980, Turkey upgraded all its legations to embassies with ambassadors, including Israel. Up until then, you had to be a close ally to get an ambassador. So technically, the first Turkish ambassador did not arrive until 1980, but diplomacy was normalized in 1950.
When Erdogan came to power in 2002, he immediate made relations closer. He referred to Israel as the closest non-NATO ally, and made a huge state visit in 2005. In 2006, Israeli Foreign Ministry referred to relations with Turkey by one word, "perfect", which was closer than even relations with the USA.
In 2009, things began to fall apart. By 2011, diplomatic relations were suspended. In 2016, they were unsuspended, and relations were normalized, again. Ambassadors were again exchanged. Later in 2018, ambassadors were withdrawn, but embassies remained. Now ambassadors are returning.